{"id":1114,"date":"2021-10-13T17:07:40","date_gmt":"2021-10-13T17:07:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1114"},"modified":"2026-03-30T17:18:17","modified_gmt":"2026-03-30T17:18:17","slug":"1-2-4-long-division-of-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-2-4-long-division-of-integers\/","title":{"raw":"1.2.3: Long Division of Integers","rendered":"1.2.3: Long Division of Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Use division notation<\/li>\r\n \t<li>Divide and check the solution using multiplication<\/li>\r\n \t<li>Identify and use the division properties of one<\/li>\r\n \t<li>Identify and use the division properties of zero<\/li>\r\n \t<li>Use the long division algorithm to divide multiple-digit integers<\/li>\r\n \t<li>Identify the divisor, dividend, and remainder in a division problem<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Quotient<\/strong>: the result of dividing two integers<\/li>\r\n \t<li><strong>Dividend<\/strong>: the integer being divided<\/li>\r\n \t<li><strong>Divisor<\/strong>: the integer dividing the dividend<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Use Division Notation<\/h2>\r\nSuppose you have [latex]12[\/latex] cookies and want to package them in bags with [latex]4[\/latex] cookies in each bag. How many bags would we need?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215605\/CNX_BMath_Figure_01_05_001.png\" alt=\"Three rows of four cookies to show twelve cookies.\" width=\"105\" height=\"77\" \/>\r\nYou might put [latex]4[\/latex] cookies in the first bag, [latex]4[\/latex] in the second bag, and so on, until you run out of cookies. Doing it this way, you would fill [latex]3[\/latex] bags.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215607\/CNX_BMath_Figure_01_05_042.png\" alt=\"Three bags of cookies, each bag containing 4 cookies.\" width=\"314\" height=\"141\" \/>\r\n\r\nWe can write\r\n<p style=\"text-align: center;\">[latex]12\\div 4[\/latex]<\/p>\r\nWe read this as <em>twelve divided by four<\/em> and the result of\u00a0<span style=\"font-size: 1em;\">latex]3[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0is the <\/span><em><strong style=\"font-size: 1rem; text-align: initial;\">quotient<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\"> of [latex]12[\/latex] and [latex]4[\/latex]. \u00a0We call the number being divided, <\/span>the <em><strong style=\"font-size: 1rem; text-align: initial;\">dividend<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">, and the number dividing it,\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">the <em><strong>divisor<\/strong><\/em>. In this case, the dividend is [latex]12[\/latex] and the divisor is [latex]4[\/latex].<\/span>\r\n\r\nThere are four different ways of notating division, but they all mean the same thing: \u00a0[latex]4\\overline{)12}[\/latex], [latex]12\\div 4, 12\\text{\/}4, \\frac{12}{4}[\/latex]. In each case, the [latex]12[\/latex] is the <strong>dividend<\/strong> and the [latex]4[\/latex] is the <strong>divisor<\/strong>.\r\n<h2>Dividing Integers<\/h2>\r\nDivision is the inverse operation of multiplication. This means that division undoes multiplication. We know [latex]12\\div 4=3[\/latex] because [latex]3\\cdot 4=12[\/latex]. Knowing all the multiplication number facts is very important when doing division.\r\n\r\nWe check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know [latex]24\\div 8=3[\/latex] is correct because [latex]3\\cdot 8=24[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide. Then check by multiplying.\r\n<ol>\r\n \t<li>[latex]-42\\div 6[\/latex]<\/li>\r\n \t<li>[latex]\\frac{-72}{-9}[\/latex]<\/li>\r\n \t<li>[latex]7\\overline{)63}[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168287031935\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-42\\div 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]-42[\/latex] by [latex]6[\/latex].<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]-7\\cdot 6[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-42\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id11682870335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{-72}{-9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]-72[\/latex] by [latex]-9[\/latex].<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]8\\cdot (-9)[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-72\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id11670335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<th>3.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7\\overline{)63}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]63[\/latex] by [latex]7[\/latex].<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]9\\cdot 7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]63\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen the divisor or the dividend has more than one digit, it is usually easier to use the [latex]4\\overline{)12}[\/latex] notation. This process is called <em><strong>long division<\/strong><\/em>. Let\u2019s work through the process by dividing [latex]78[\/latex] by [latex]3[\/latex].\r\n<table id=\"eip-244\" class=\"unnumbered unstyled\" style=\"width: 970.438px;\" summary=\"This image has 2 columns. the left column contains instructions and the right column contains expressions. The exercises being worked out is 78 divided by 3. The first line reads: Divide the first digit of dividend 7, by the divisor, 3. The next line reads: the divisor 3 can go into 7 two times since 2 times 3 equals 6. Write the 2 above the 7 in the quotient. Next to this shows the expression 3 divided by 78, with the two above the seven in the quotient. The next line reads \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 599px;\">Dividing [latex]78[\/latex] by [latex]3[\/latex]<\/th>\r\n<td style=\"width: 337px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Divide the first digit of the dividend, [latex]7[\/latex], by the divisor, [latex]3[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">The divisor [latex]3[\/latex] can go into [latex]7[\/latex] two times, since [latex]2\\times 3=6[\/latex] . Write the [latex]2[\/latex] above the [latex]7[\/latex] in the quotient.<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215615\/CNX_BMath_Figure_01_05_043_img-02.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is placed above the 7 in the dividend. \" width=\"38\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Multiply the [latex]2[\/latex] in the quotient by [latex]2[\/latex] and write the product, [latex]6[\/latex], under the [latex]7[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-03.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend.\" width=\"35\" height=\"64\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Subtract that product from the first digit in the dividend. Subtract [latex]7 - 6[\/latex] . Write the difference, 1, under the first digit in the dividend.<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-04.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6.\" width=\"35\" height=\"78\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Bring down the next digit of the dividend. Bring down the [latex]8[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-05.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6. The number 8 is written to the right of the number 1. \" width=\"35\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Divide [latex]18[\/latex] by the divisor, [latex]3[\/latex]. The divisor [latex]3[\/latex] goes into [latex]18[\/latex] six times.<\/td>\r\n<td style=\"width: 337px;\" rowspan=\"2\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-06.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 26 is written above the dividend 78 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6. The number 8 is written to the right of the number 1. \" width=\"35\" height=\"80\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Write [latex]6[\/latex] in the quotient above the [latex]8[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor and write the product, [latex]18[\/latex], under the dividend. Subtract [latex]18[\/latex] from [latex]18[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215618\/CNX_BMath_Figure_01_05_043_img-07.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 26 is written above the dividend 78 and the number 6 is written below the 7 in the dividend. The number 18 is written below the number 6. Another number 18 is written below the number 18, then a number 0 is written below the number 18. \" width=\"35\" height=\"117\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.\r\n<p style=\"text-align: center;\">[latex]\\text{So }78\\div 3=26[\/latex].<\/p>\r\nCheck by multiplying the quotient times the divisor to get the dividend. Multiply [latex]26\\times 3[\/latex] to make sure that product equals the dividend, [latex]78[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill \\stackrel{1}{2}6\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\\\ \\hfill 78 \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">It does, so our answer is correct.\u00a0[latex]\\checkmark[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Divide INTEGERS USING LONG DIVISION<\/h3>\r\n<ol id=\"eip-id1168288534169\" class=\"stepwise\">\r\n \t<li>Determine the sign of the quotient, then work with the absolute values of the dividend and divisor.<\/li>\r\n \t<li>Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.<\/li>\r\n \t<li>Write the quotient above the dividend.<\/li>\r\n \t<li>Multiply the quotient by the divisor and write the product under the dividend.<\/li>\r\n \t<li>Subtract that product from the dividend.<\/li>\r\n \t<li>Bring down the next digit of the dividend.<\/li>\r\n \t<li>Repeat from Step 1 until there are no more digits in the dividend to bring down.<\/li>\r\n \t<li>Check by multiplying the quotient times the divisor.<\/li>\r\n \t<li>Add the sign of division to the quotient of the absolute values.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the video below we show another example of using long division.\r\n\r\nhttps:\/\/youtu.be\/KvVhaB5mqr8\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]-2,596\\div 4[\/latex]. Check by multiplying:\r\n[reveal-answer q=\"252445\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"252445\"]\r\n\r\nSolution\r\n<table id=\"eip-287\" class=\"unnumbered unstyled\" style=\"width: 917px; height: 730px;\" summary=\"This image shows long division for solving the expression 4 divided by 2,596. The first line says to divide the first digit of the dividend, 2, by the divisor, 4. The expression shows the 2 in red. The next line says that since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. the divisor 4 goes into 25 six times. We write the six in the quotient above the 5. The expression now shows the 6 above the 5 in the quotient. Next, multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. The expression shows the 24 under the 25. Subtract 25 minus 4. Write the difference, 1, under the second digit in the dividend. The expression shows in long division, 25 minus 24 equals 1. Now bring the 9 down and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. The expression now shows in long division, the 9 in the dividend brought down next to the 1, to make 19, and the 4 in the quotient above the 9. Under the 19 is 16 with a difference of 3. The next line says to bring down the 6 and repeat these steps. there are 9 fours in 36. Write 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. The expression shows in long division the 6 brought down next to the 3 to make 36. The 9 in the quotient above the 6 for an answer of 649.\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 675.5px;\">Divide [latex]-2,596\\div 4[\/latex].<\/th>\r\n<td style=\"width: 217.828125px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 675.5px; height: 12px;\">Decide on the sign of the quotient.<\/td>\r\n<td style=\"width: 217.828125px; height: 12px;\">(-)\/(+) = (-)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 29px;\">\r\n<td style=\"width: 675.5px; height: 29px;\">Let's rewrite the problem to set it up for long division using the absolute values.<\/td>\r\n<td style=\"width: 217.828125px; height: 29px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215619\/CNX_BMath_Figure_01_05_044_img-01.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596.\" width=\"84\" height=\"27\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 29px;\">\r\n<td style=\"width: 675.5px; height: 29px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\r\n<td style=\"width: 217.828125px; height: 29px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-02.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596.\" width=\"84\" height=\"27\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"width: 675.5px; height: 24px;\">Since [latex]4[\/latex] does not go into [latex]2[\/latex], we use the first two digits of the dividend and divide [latex]25[\/latex] by [latex]4[\/latex]. The divisor [latex]4[\/latex] goes into [latex]25[\/latex] six times.<\/td>\r\n<td style=\"width: 217.828125px; height: 24px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"width: 675.5px; height: 48px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\r\n<td style=\"width: 217.828125px; height: 48px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-03.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend. \" width=\"84\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 66px;\">\r\n<td style=\"width: 675.5px; height: 66px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor [latex]4[\/latex] and write the product, [latex]24[\/latex], under the first two digits in the dividend.<\/td>\r\n<td style=\"width: 217.828125px; height: 66px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215621\/CNX_BMath_Figure_01_05_044_img-04.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend. The number 24 is written below the 25 in the dividend.\" width=\"84\" height=\"64\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 87px;\">\r\n<td style=\"width: 675.5px; height: 87px;\">Subtract that product from the first two digits in the dividend. Subtract [latex]25 - 24[\/latex] . Write the difference, [latex]1[\/latex], under the second digit in the dividend.<\/td>\r\n<td style=\"width: 217.828125px; height: 87px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215622\/CNX_BMath_Figure_01_05_044_img-05.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24. \" width=\"84\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 127px;\">\r\n<td style=\"width: 675.5px; height: 127px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]4[\/latex] fours in [latex]19[\/latex]. Write the [latex]4[\/latex] over the [latex]9[\/latex]. Multiply the [latex]4[\/latex] by [latex]4[\/latex] and subtract this product from [latex]19[\/latex].<\/td>\r\n<td style=\"width: 217.828125px; height: 127px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215624\/CNX_BMath_Figure_01_05_044_img-06.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 64 is written above the 59 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24. The number 9 in the dividend is written again to the right of the number 1. The number 16 is written below the number 19, and then the number 3 is written below the 6 of 16. \" width=\"84\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 163px;\">\r\n<td style=\"width: 675.5px; height: 163px;\">Bring down the [latex]6[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]36[\/latex]. Write the [latex]9[\/latex] over the [latex]6[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]36[\/latex].<\/td>\r\n<td style=\"width: 217.828125px; height: 163px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215625\/CNX_BMath_Figure_01_05_044_img-07.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 649 is written above the 596 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24. The number 9 in the dividend is written again to the right of the number 1. The number 16 is written below the number 19, and then the number 3 is written below the 6 of 16. The number 6 in the dividend is written again to the right of the number 3. The number 36 is written under the number 36. The number 0 is written below the number 6 of 36. \" width=\"84\" height=\"161\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 217.828125px; height: 12px;\"><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr style=\"height: 133px;\">\r\n<td style=\"width: 675.5px; height: 133px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215626\/CNX_BMath_Figure_01_05_044_img-08.png\" alt=\"The number 4 is placed below the 9 of the number 649. A multiplication symbol is placed to the far left of the number 4. the number 2596 is placed below the number 4 where the 6 of 2596 is right below the number 4. \" width=\"84\" height=\"84\" \/><\/td>\r\n<td style=\"width: 217.828125px; height: 133px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our arithmetic is correct.\r\n\r\nAdd the sign of the answer from the first step:\u00a0[latex]-2,596\\div 4=-649[\/latex] .\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144636[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]4,506\\div 6[\/latex]. Check by multiplying:\r\n[reveal-answer q=\"474096\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474096\"]\r\n\r\nSolution\r\n<table id=\"eip-483\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 4,506 divided by six is worked out in long division. The first line reads first we try to divide 6 into 4. Since that won't work, we try 6 into 45. There are 7 sixes in 45. We write the 7 over the 5. The expression shows the 7 above the 5 in the quotient. Multiply the 7 by the 6 and subtract the product from 45. The expression shows in long division, 45 minus 42, equals 3. The next line reads \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 672.847px;\">Divide [latex]4,506\\div 6[\/latex].<\/th>\r\n<td style=\"width: 222.153px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215627\/CNX_BMath_Figure_01_05_045_img-01.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506.\" width=\"84\" height=\"32\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">First we try to divide [latex]6[\/latex] into [latex]4[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-02.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506.\" width=\"84\" height=\"27\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Since that won't work, we try [latex]6[\/latex] into [latex]45[\/latex].\r\n\r\nThere are [latex]7[\/latex] sixes in [latex]45[\/latex]. We write the [latex]7[\/latex] over the [latex]5[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-03.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 7 is placed above the 5 in the dividend.\" width=\"84\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Multiply the [latex]7[\/latex] by [latex]6[\/latex] and subtract this product from [latex]45[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215629\/CNX_BMath_Figure_01_05_045_img-04.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 7 is placed above the 5 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42. \" width=\"84\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Now bring down the [latex]0[\/latex] and repeat these steps. There are [latex]5[\/latex] sixes in [latex]30[\/latex]. Write the [latex]5[\/latex] over the [latex]0[\/latex]. Multiply the [latex]5[\/latex] by [latex]6[\/latex] and subtract this product from [latex]30[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215631\/CNX_BMath_Figure_01_05_045_img-05.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 75 is placed above the 50 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42. The 0 in the dividend is written again to the right of the number 3. Another number 30 is written below the number 30. The number 0 is written below the 0 of 30. \" width=\"84\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] six in [latex]6[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply [latex]1[\/latex] by [latex]6[\/latex] and subtract this product from [latex]6[\/latex]<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215632\/CNX_BMath_Figure_01_05_045_img-06.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 751 is placed above the 506 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42. The 0 in the dividend is written again to the right of the number 3. Another number 30 is written below the number 30. The number 0 is written below the 0 of 30. The 6 in the dividend is written again next to the number 0. Another number 6 is written below the number 6. The number 0 is written below the number 6.\" width=\"84\" height=\"164\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_045_img-07.png\" alt=\"The number 6 is placed below the 1 of the number 751. A multiplication symbol is placed to the far left of the number 6. The number 4506 is placed below the number 6 where the 6 in 4506 is right below the number 6. \" width=\"84\" height=\"78\" \/><\/td>\r\n<td style=\"width: 222.153px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our answer is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144640[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]7,263\\div -9[\/latex]. Check by multiplying.\r\n[reveal-answer q=\"483321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"483321\"]\r\n\r\nSolution\r\n<table id=\"eip-485\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 7,263 divided by 9 is worked out in long division. The first line states \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 717.326px;\">Divide [latex]7,263\\div -9[\/latex].<\/th>\r\n<td style=\"width: 177.674px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Decide on the sign of the quotient.<\/td>\r\n<td style=\"width: 177.674px;\">\u00a0(+)\/(-) = (-)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Let's rewrite the problem to set it up for long division using the absolute values.<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_046_img-01.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. \" width=\"84\" height=\"30\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">First we try to divide [latex]9[\/latex] into [latex]7[\/latex].<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215634\/CNX_BMath_Figure_01_05_046_img-02.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263.\" width=\"84\" height=\"29\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Since that won't work, we try [latex]9[\/latex] into [latex]72[\/latex]. There are [latex]8[\/latex] nines in [latex]72[\/latex]. We write the [latex]8[\/latex] over the [latex]2[\/latex].<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215635\/CNX_BMath_Figure_01_05_046_img-03.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 8 is placed above the 2 in the dividend. \" width=\"84\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Multiply the [latex]8[\/latex] by [latex]9[\/latex] and subtract this product from [latex]72[\/latex].<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215636\/CNX_BMath_Figure_01_05_046_img-04.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 8 is placed above the 2 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72. \" width=\"84\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There are [latex]0[\/latex] nines in [latex]6[\/latex]. Write the [latex]0[\/latex] over the [latex]6[\/latex]. Multiply the [latex]0[\/latex] by [latex]9[\/latex] and subtract this product from [latex]6[\/latex].<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215637\/CNX_BMath_Figure_01_05_046_img-05.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 80 is placed above the 26 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72. The 6 in the dividend is written again to the right of the number 0. Another number 0 is written below the number 6. Another number 6 is written below the number 0.\" width=\"84\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]7[\/latex] nines in [latex]63[\/latex]. Write the [latex]7[\/latex] over the [latex]3[\/latex]. Multiply the [latex]7[\/latex] by [latex]9[\/latex] and subtract this product from [latex]63[\/latex].<\/td>\r\n<td style=\"width: 177.674px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215637\/CNX_BMath_Figure_01_05_046_img-06.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 807 is placed above the 263 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72. The 6 in the dividend is written again to the right of the number 0. Another number 0 is written below the number 6. Another number 6 is written below the number 0. The 3 in the dividend is written again next to the right of the number 6. Another number 63 is written below the number 63. The number 0 is written below the 3 of the number 63.\" width=\"84\" height=\"164\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 717.326px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215638\/CNX_BMath_Figure_01_05_046_img-07.png\" alt=\"The number 9 is placed below the 7 of the number 751. A multiplication symbol is placed to the far left of the number 9. The number 7263 is placed below the number 9 where the 3 in 7263 is right below the number 9. \" width=\"84\" height=\"78\" \/><\/td>\r\n<td style=\"width: 177.674px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our arithmetic is correct.\r\n\r\nAdd the sign of the answer from the first step:\u00a0\u00a0[latex]7,263\\div -9= -807[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nDivide using long division: [latex]-764\\div 4[\/latex]\r\n\r\n[reveal-answer q=\"H0022\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H0022\"]\r\n\r\n[latex]-191[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch this video for another example of how to use long division to divide a four-digit integer by a two-digit integer.\r\n\r\nhttps:\/\/youtu.be\/V7Korf09iWI\r\n\r\nSo far, all the division problems have worked out evenly. For example, if we had [latex]24[\/latex] cookies and wanted to make bags of [latex]8[\/latex] cookies, we would have [latex]3[\/latex] bags. But what if there were [latex]28[\/latex] cookies and we wanted to make bags of [latex]8?[\/latex] Start with the [latex]28[\/latex] cookies.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215640\/CNX_BMath_Figure_01_05_027.png\" alt=\"28 cookies placed at random.\" width=\"316\" height=\"127\" \/>\r\nTry to put the cookies in groups of eight.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215642\/CNX_BMath_Figure_01_05_028.png\" alt=\"There are 28 cookies where three groups are circled with 8 cookies in each group, leaving 3 cookies outside the three circles.\" width=\"319\" height=\"141\" \/>\r\nThere are [latex]3[\/latex] groups of eight cookies, and [latex]4[\/latex] cookies left over. We call the [latex]4[\/latex] cookies that are left over the remainder and show it by writing R4 next to the [latex]3[\/latex]. (The R stands for <em><strong>remainder<\/strong><\/em>).\r\n\r\nTo check this division, we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then add the remainder of [latex]4[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 3\\\\ \\hfill \\underset{\\text{___}}{\\times 8}\\\\ \\hfill 24\\\\ \\hfill \\underset{\\text{___}}{+4}\\\\ \\hfill 28\\end{array}[\/latex]<\/p>\r\nThis is an example of the <em><strong>Fundamental Theorem of Arithmetic<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>The Fundamental Theorem OF Arithmetic<\/h3>\r\n<p style=\"text-align: center;\"><span class=\"TextRun SCXW89433696 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"none\"><span class=\"NormalTextRun SCXW89433696 BCX0\">Number = Divisor \u00b7 Quotient + Remainder<\/span><\/span><span class=\"EOP SCXW89433696 BCX0\" data-ccp-props=\"{&quot;134233279&quot;:true}\">\u00a0<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]1,439\\div 4[\/latex]. Check by multiplying.\r\n[reveal-answer q=\"498101\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"498101\"]\r\n\r\nSolution\r\n<table id=\"eip-879\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 1,439 divided by 4. The first line says to try to divide 4 into 1. Since this won't work, try 4 into 14. There are 3 fours in 14. Write the 3 over the 4. the expression shows the quotient 3 above the 4. Next, multiply the 3 by the 4 and subtract this product from 14. The expressions shows this in long division with the quotient 5 above the 3. The next line says to bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract the product from 23. The expression shows this in long division. The next line says to bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract the product from 39. There are no more numbers to bring down. There is a remainder of 3. The expression shows the answer of 359 remainder 3.\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 695.851px;\">Divide [latex]1,439\\div 4[\/latex].<\/th>\r\n<td style=\"width: 199.149px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215643\/CNX_BMath_Figure_01_05_047_img-01.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439.\" width=\"171\" height=\"28\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">First we try to divide [latex]4[\/latex] into [latex]1[\/latex]. Since that won't work, we try [latex]4[\/latex] into [latex]14[\/latex]. There are [latex]3[\/latex] fours in [latex]14[\/latex]. We write the [latex]3[\/latex] over the [latex]4[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215644\/CNX_BMath_Figure_01_05_047_img-02.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 3 is placed above the 4 in the dividend. \" width=\"171\" height=\"45\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Multiply the [latex]3[\/latex] by [latex]4[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215645\/CNX_BMath_Figure_01_05_047_img-03.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 3 is placed above the 4 in the dividend. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12.\" width=\"171\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]5[\/latex] fours in [latex]23[\/latex]. Write the [latex]5[\/latex] over the [latex]3[\/latex]. Multiply the [latex]5[\/latex] by [latex]4[\/latex] and subtract this product from [latex]23[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215646\/CNX_BMath_Figure_01_05_047_img-04.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 35 is placed above the 43 in the dividend. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12. The 3 in the dividend is written again to the right of the number 2. Another number 20 is written below the number 23. Another number 3 is written below the number 0in 20. \" width=\"171\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]39[\/latex]. Write the [latex]9[\/latex] over the [latex]9[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]39[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]3[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215647\/CNX_BMath_Figure_01_05_047_img-05.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 359 is placed above the 439 in the dividend. The phrase &quot;R3&quot; is placed to the right of 359. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12. The 3 in the dividend is written again to the right of the number 2. Another number 20 is written below the number 23. Another number 3 is written below the number 0in 20. The 9 in the dividend is written again next to the right of the number 3. Another number 36 is written below the number 39. The number 3 is written below the 6 of the number 36.\" width=\"171\" height=\"164\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215648\/CNX_BMath_Figure_01_05_047_img-06.png\" alt=\"The number 4 is placed below the 9 of the number 359. A multiplication symbol is placed far to the left of the number 4. The number 1436 is placed below the number 4 where the 6 in 1436 is right below the number 4. The number 3 is below the 6 in 1436. A addition symbol is placed far to the left of the number 3. The number 1439 is placed below the number 3, where 9 in 1439 is right below 6 in 1436. \" width=\"171\" height=\"125\" \/><\/td>\r\n<td style=\"width: 199.149px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]1,439\\div 4[\/latex] is [latex]359[\/latex] with a remainder of [latex]3[\/latex]. Our answer is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144643[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide and then check by multiplying: [latex]-1,461\\div (-13)[\/latex].\r\n[reveal-answer q=\"174689\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"174689\"]\r\n\r\nSolution\r\n<table id=\"eip-708\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\"This image shows 1,461 divided by 13. The first line says First we try to divide 13 into 1. Since that won\u2019t work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4. Multiply the 1 by 13 and subtract this product from 14. The expression shows the quotient 1 above the 4, and 13 under 14 with the difference as 1. The next line says to bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16. The expression shows the 6 in the dividend with a blue arrow pointing downward to show the 6 is brought down next to the 1. In long division, 16 minus 3 shows a difference of 3. Next it says, now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. The expression shows in long division the 1 in the dividend with a blue arrow pointing downward to show the one brought down to make 31. A 2 is placed above the 1 in the dividend. Thirty-one minus 26 is 5, but there are no more numbers to bring down, so we are done. The remainder is 5. The answer is 112 remainder 5. The next line says to check by multiplying in which the expression shows 112 with the word \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 701.698px;\">[latex]-1,461\\div (-13)[\/latex].<\/th>\r\n<td style=\"width: 192.302px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Decide on the sign of the quotient.<\/td>\r\n<td style=\"width: 192.302px;\">(-)\/(-) = (+)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 192.302px;\">[latex]13\\overline{)1,461}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">First we try to divide [latex]13[\/latex] into [latex]1[\/latex]. Since that won't work, we try [latex]13[\/latex] into [latex]14[\/latex]. There is [latex]1[\/latex] thirteen in [latex]14[\/latex]. We write the [latex]1[\/latex] over the [latex]4[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215649\/CNX_BMath_Figure_01_05_048_img-02.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 1 is placed above the 4 in the dividend. \" width=\"173\" height=\"43\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215650\/CNX_BMath_Figure_01_05_048_img-03.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 1 is placed above the 4 in the dividend. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13. \" width=\"173\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] thirteen in [latex]16[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]16[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215651\/CNX_BMath_Figure_01_05_048_img-04.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 11 is placed above the 46 in the dividend. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13. The 6 in the dividend is written again to the right of the number 1. Another number 13 is written below the number 16. Another number 3 is written below the number 3 in 13. \" width=\"173\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Now bring down the [latex]1[\/latex] and repeat these steps. There are [latex]2[\/latex] thirteens in [latex]31[\/latex]. Write the [latex]2[\/latex] over the [latex]1[\/latex]. Multiply the [latex]2[\/latex] by [latex]13[\/latex] and subtract this product from [latex]31[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]5[\/latex]. [latex]1,462\\div 13 [\/latex] is [latex]112[\/latex] with a remainder of [latex]5[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215652\/CNX_BMath_Figure_01_05_048_img-05.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 112 is placed above the 461 in the dividend. The phrase &quot;R5&quot; is placed to the right of 112. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13. The 6 in the dividend is written again to the right of the number 1. Another number 13 is written below the number 16. Another number 3 is written below the number 3 in 13. The 1 in the dividend is written again next to the right of the number 3. Another number 26 is written below the number 31. The number 5 is written below the 6 of the number 26.\" width=\"173\" height=\"164\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215653\/CNX_BMath_Figure_01_05_048_img-06.png\" alt=\"The number 13 (divisor) is placed below the 12 of the number 112 (quotient). A multiplication symbol is placed far to the left of the number 13. The number 336 is placed below the number 13 where the 6 in 336 is right below the number 3 in 13. The number 1120 is placed below the number 336 where 0 in 1120 is right below the 6 in 336. Another number 5 (remainder) is placed below the number 1120. A addition symbol is placed far to the left of the number 5. The number 1461 is placed below the number 5, where 1 in 1461 is right below the number 5. \" width=\"173\" height=\"136\" \/><\/td>\r\n<td style=\"width: 192.302px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our arithmetic is correct.\r\n\r\nAdd the sign of the answer from the first step:\u00a0[latex]-1,461\\div (-13)=+(112\\text{R}5)= 112\\text{R}5[\/latex] Notice that when the quotient is positive, so is the remainder.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144644[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide and check by multiplying: [latex]-74,521\\div 241[\/latex].\r\n[reveal-answer q=\"862075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862075\"]\r\n\r\nSolution\r\n<table id=\"eip-787\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 74,521 divided by 241. First we try to divide 241 into 7. Since that won\u2019t work, we try 241 into 74. That still won\u2019t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3. Since 3 times 241 equals 723, we write the 3 over the 5 in 745. The image shows the expression in long division with the quotient 3 above the 5. Next it says to bring down the 2 and repeat these steps. 241 does not divide into 222. Write a 0 over the 2 as a place holder and then continue. Note that 4 would be too large because 4 times 241 equals 964, which is greater than 745. Multiply the 3 by 241 and subtract this product from 745. The image shows the expression in long division 745 minus 723, which is 22. Next, it says to bring down the 2 and repeat these steps. 241 does not divide into 222. Write a 0 over the 2 as a place holder and then continue. Next, it says to bring down the 1 and repeat these steps. Try 9 Since 9 times 241 equals 2,169, we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221. The expression shows the one brought down and the subtraction of 2,221 minus 2,169 which gives the answer as 309 remainder 52. The image is checked through multiplication. The numbers 309 times 241are multiplied vertically. The expression is worked out to show that 74,3521 checks and the answer is correct.\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 692.424px;\">[latex]-74,521\\div 241[\/latex].<\/th>\r\n<td style=\"width: 202.576px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Decide on the sign of the quotient.<\/td>\r\n<td style=\"width: 202.576px;\">(-)\/(+) = (-)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 202.576px;\">[latex]241\\overline{)74,521}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">First we try to divide [latex]241[\/latex] into [latex]7[\/latex]. Since that won\u2019t work, we try [latex]241[\/latex] into [latex]74[\/latex]. That still won\u2019t work, so we try [latex]241[\/latex] into[latex]745[\/latex]. Since [latex]2[\/latex] divides into [latex]7[\/latex] three times, we try [latex]3[\/latex]. Since [latex]3\\times 241=723[\/latex] , we write the [latex]3[\/latex] over the [latex]5[\/latex] in [latex]745[\/latex]. Note that [latex]4[\/latex] would be too large, because [latex]4\\times 241=964[\/latex] , which is greater than [latex]745[\/latex].<\/td>\r\n<td style=\"width: 202.576px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Multiply the [latex]3[\/latex] by [latex]241[\/latex] and subtract this product from [latex]745[\/latex].<\/td>\r\n<td style=\"width: 202.576px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215654\/CNX_BMath_Figure_01_05_049_img-02.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 3 is placed above the 5 in the dividend. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723. \" width=\"175\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Now bring down the [latex]2[\/latex] and repeat these steps. [latex]241[\/latex] does not divide into [latex]222[\/latex].\r\n\r\nWe write a [latex]0[\/latex] over the [latex]2[\/latex] as a placeholder and then continue.<\/td>\r\n<td style=\"width: 202.576px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215655\/CNX_BMath_Figure_01_05_049_img-03.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 30 is placed above the 52 in the dividend. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723. The 2 in the dividend is written again to the right of the number 22. \" width=\"175\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Now bring down the [latex]1[\/latex] and repeat these steps. Try [latex]9[\/latex]. Since [latex]9\\times 241=2,169[\/latex] , we write the [latex]9[\/latex] over the [latex]1[\/latex]. Multiply the [latex]9[\/latex] by [latex]241[\/latex] and subtract this product from [latex]2,221[\/latex].<\/td>\r\n<td style=\"width: 202.576px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215656\/CNX_BMath_Figure_01_05_049_img-04.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 309 is placed above the 521 in the dividend. The phrase &quot;R52&quot; is placed to the right of 309. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723. The 21 in the dividend is written again to the right of the number 22. Another number 2169 is written below the number 2221. Another number 52 is written below 69 in 2169. \" width=\"175\" height=\"125\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">There are no more numbers to bring down, so we are finished. The remainder is [latex]52[\/latex]. So [latex]74,521\\div 241[\/latex] is [latex]309[\/latex] with a remainder of [latex]52[\/latex].<\/td>\r\n<td style=\"width: 202.576px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 692.424px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215657\/CNX_BMath_Figure_01_05_049_img-05.png\" alt=\"The number 241 (divisor) is placed below the number 309 (quotient). A multiplication symbol is placed far to the left of the number 241. The number 309 is placed below the number 241. The number 12360 is placed below the number 309. The number 61800 is placed below the number 12360. Another number 74469 is placed below the number 61800. Another number 52 (remainder) is placed below the number 72469. A addition symbol is placed far to the left of the number 52. The number 74521 is placed below the number 52.\" width=\"167\" height=\"198\" \/><\/td>\r\n<td style=\"width: 202.576px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our arithmetic is correct.\r\n\r\nAdd the sign of the answer from the first step:\u00a0 [latex]-74,521\\div 241= -\\left(309\\text{R}52\\right)[\/latex]. When the quotient is negative, so is the remainder, so this simplifies to\u00a0[latex]-74,521\\div 241= -309\\text{R}(-52)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nDivide 8452 by \u20139.\r\n\r\n[reveal-answer q=\"910292\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"910292\"]\u2013939 R(1)[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the video below for another example of how to use long division to divide integers when there is a remainder.\r\n\r\nhttps:\/\/youtu.be\/UPUcShGCBOs\r\n\r\n&nbsp;\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Use division notation<\/li>\n<li>Divide and check the solution using multiplication<\/li>\n<li>Identify and use the division properties of one<\/li>\n<li>Identify and use the division properties of zero<\/li>\n<li>Use the long division algorithm to divide multiple-digit integers<\/li>\n<li>Identify the divisor, dividend, and remainder in a division problem<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Quotient<\/strong>: the result of dividing two integers<\/li>\n<li><strong>Dividend<\/strong>: the integer being divided<\/li>\n<li><strong>Divisor<\/strong>: the integer dividing the dividend<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Use Division Notation<\/h2>\n<p>Suppose you have [latex]12[\/latex] cookies and want to package them in bags with [latex]4[\/latex] cookies in each bag. How many bags would we need?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215605\/CNX_BMath_Figure_01_05_001.png\" alt=\"Three rows of four cookies to show twelve cookies.\" width=\"105\" height=\"77\" \/><br \/>\nYou might put [latex]4[\/latex] cookies in the first bag, [latex]4[\/latex] in the second bag, and so on, until you run out of cookies. Doing it this way, you would fill [latex]3[\/latex] bags.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215607\/CNX_BMath_Figure_01_05_042.png\" alt=\"Three bags of cookies, each bag containing 4 cookies.\" width=\"314\" height=\"141\" \/><\/p>\n<p>We can write<\/p>\n<p style=\"text-align: center;\">[latex]12\\div 4[\/latex]<\/p>\n<p>We read this as <em>twelve divided by four<\/em> and the result of\u00a0<span style=\"font-size: 1em;\">latex]3[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0is the <\/span><em><strong style=\"font-size: 1rem; text-align: initial;\">quotient<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\"> of [latex]12[\/latex] and [latex]4[\/latex]. \u00a0We call the number being divided, <\/span>the <em><strong style=\"font-size: 1rem; text-align: initial;\">dividend<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">, and the number dividing it,\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">the <em><strong>divisor<\/strong><\/em>. In this case, the dividend is [latex]12[\/latex] and the divisor is [latex]4[\/latex].<\/span><\/p>\n<p>There are four different ways of notating division, but they all mean the same thing: \u00a0[latex]4\\overline{)12}[\/latex], [latex]12\\div 4, 12\\text{\/}4, \\frac{12}{4}[\/latex]. In each case, the [latex]12[\/latex] is the <strong>dividend<\/strong> and the [latex]4[\/latex] is the <strong>divisor<\/strong>.<\/p>\n<h2>Dividing Integers<\/h2>\n<p>Division is the inverse operation of multiplication. This means that division undoes multiplication. We know [latex]12\\div 4=3[\/latex] because [latex]3\\cdot 4=12[\/latex]. Knowing all the multiplication number facts is very important when doing division.<\/p>\n<p>We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know [latex]24\\div 8=3[\/latex] is correct because [latex]3\\cdot 8=24[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Then check by multiplying.<\/p>\n<ol>\n<li>[latex]-42\\div 6[\/latex]<\/li>\n<li>[latex]\\frac{-72}{-9}[\/latex]<\/li>\n<li>[latex]7\\overline{)63}[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168287031935\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-42\\div 6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]-42[\/latex] by [latex]6[\/latex].<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]-7\\cdot 6[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-42\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id11682870335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{-72}{-9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]-72[\/latex] by [latex]-9[\/latex].<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]8\\cdot (-9)[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-72\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id11670335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\n<tbody>\n<tr>\n<th>3.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]7\\overline{)63}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]63[\/latex] by [latex]7[\/latex].<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]9\\cdot 7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]63\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When the divisor or the dividend has more than one digit, it is usually easier to use the [latex]4\\overline{)12}[\/latex] notation. This process is called <em><strong>long division<\/strong><\/em>. Let\u2019s work through the process by dividing [latex]78[\/latex] by [latex]3[\/latex].<\/p>\n<table id=\"eip-244\" class=\"unnumbered unstyled\" style=\"width: 970.438px;\" summary=\"This image has 2 columns. the left column contains instructions and the right column contains expressions. The exercises being worked out is 78 divided by 3. The first line reads: Divide the first digit of dividend 7, by the divisor, 3. The next line reads: the divisor 3 can go into 7 two times since 2 times 3 equals 6. Write the 2 above the 7 in the quotient. Next to this shows the expression 3 divided by 78, with the two above the seven in the quotient. The next line reads\">\n<tbody>\n<tr>\n<th style=\"width: 599px;\">Dividing [latex]78[\/latex] by [latex]3[\/latex]<\/th>\n<td style=\"width: 337px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Divide the first digit of the dividend, [latex]7[\/latex], by the divisor, [latex]3[\/latex].<\/td>\n<td style=\"width: 337px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">The divisor [latex]3[\/latex] can go into [latex]7[\/latex] two times, since [latex]2\\times 3=6[\/latex] . Write the [latex]2[\/latex] above the [latex]7[\/latex] in the quotient.<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215615\/CNX_BMath_Figure_01_05_043_img-02.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is placed above the 7 in the dividend.\" width=\"38\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Multiply the [latex]2[\/latex] in the quotient by [latex]2[\/latex] and write the product, [latex]6[\/latex], under the [latex]7[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-03.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend.\" width=\"35\" height=\"64\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Subtract that product from the first digit in the dividend. Subtract [latex]7 - 6[\/latex] . Write the difference, 1, under the first digit in the dividend.<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-04.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6.\" width=\"35\" height=\"78\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Bring down the next digit of the dividend. Bring down the [latex]8[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-05.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 2 is written above the 7 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6. The number 8 is written to the right of the number 1.\" width=\"35\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Divide [latex]18[\/latex] by the divisor, [latex]3[\/latex]. The divisor [latex]3[\/latex] goes into [latex]18[\/latex] six times.<\/td>\n<td style=\"width: 337px;\" rowspan=\"2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-06.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 26 is written above the dividend 78 and the number 6 is written below the 7 in the dividend. The number 1 is written below the number 6. The number 8 is written to the right of the number 1.\" width=\"35\" height=\"80\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Write [latex]6[\/latex] in the quotient above the [latex]8[\/latex].<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor and write the product, [latex]18[\/latex], under the dividend. Subtract [latex]18[\/latex] from [latex]18[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215618\/CNX_BMath_Figure_01_05_043_img-07.png\" alt=\"The divisor 3 is placed to the left of the dividend 78. The quotient 26 is written above the dividend 78 and the number 6 is written below the 7 in the dividend. The number 18 is written below the number 6. Another number 18 is written below the number 18, then a number 0 is written below the number 18.\" width=\"35\" height=\"117\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{So }78\\div 3=26[\/latex].<\/p>\n<p>Check by multiplying the quotient times the divisor to get the dividend. Multiply [latex]26\\times 3[\/latex] to make sure that product equals the dividend, [latex]78[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill \\stackrel{1}{2}6\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\\\ \\hfill 78 \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">It does, so our answer is correct.\u00a0[latex]\\checkmark[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Divide INTEGERS USING LONG DIVISION<\/h3>\n<ol id=\"eip-id1168288534169\" class=\"stepwise\">\n<li>Determine the sign of the quotient, then work with the absolute values of the dividend and divisor.<\/li>\n<li>Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.<\/li>\n<li>Write the quotient above the dividend.<\/li>\n<li>Multiply the quotient by the divisor and write the product under the dividend.<\/li>\n<li>Subtract that product from the dividend.<\/li>\n<li>Bring down the next digit of the dividend.<\/li>\n<li>Repeat from Step 1 until there are no more digits in the dividend to bring down.<\/li>\n<li>Check by multiplying the quotient times the divisor.<\/li>\n<li>Add the sign of division to the quotient of the absolute values.<\/li>\n<\/ol>\n<\/div>\n<p>In the video below we show another example of using long division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Long Division - Two Digit Divided by One Digit (No Remainder)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KvVhaB5mqr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]-2,596\\div 4[\/latex]. Check by multiplying:<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q252445\">Show Solution<\/span><\/p>\n<div id=\"q252445\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-287\" class=\"unnumbered unstyled\" style=\"width: 917px; height: 730px;\" summary=\"This image shows long division for solving the expression 4 divided by 2,596. The first line says to divide the first digit of the dividend, 2, by the divisor, 4. The expression shows the 2 in red. The next line says that since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. the divisor 4 goes into 25 six times. We write the six in the quotient above the 5. The expression now shows the 6 above the 5 in the quotient. Next, multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. The expression shows the 24 under the 25. Subtract 25 minus 4. Write the difference, 1, under the second digit in the dividend. The expression shows in long division, 25 minus 24 equals 1. Now bring the 9 down and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. The expression now shows in long division, the 9 in the dividend brought down next to the 1, to make 19, and the 4 in the quotient above the 9. Under the 19 is 16 with a difference of 3. The next line says to bring down the 6 and repeat these steps. there are 9 fours in 36. Write 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. The expression shows in long division the 6 brought down next to the 3 to make 36. The 9 in the quotient above the 6 for an answer of 649.\">\n<tbody>\n<tr>\n<th style=\"width: 675.5px;\">Divide [latex]-2,596\\div 4[\/latex].<\/th>\n<td style=\"width: 217.828125px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 675.5px; height: 12px;\">Decide on the sign of the quotient.<\/td>\n<td style=\"width: 217.828125px; height: 12px;\">(-)\/(+) = (-)<\/td>\n<\/tr>\n<tr style=\"height: 29px;\">\n<td style=\"width: 675.5px; height: 29px;\">Let&#8217;s rewrite the problem to set it up for long division using the absolute values.<\/td>\n<td style=\"width: 217.828125px; height: 29px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215619\/CNX_BMath_Figure_01_05_044_img-01.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596.\" width=\"84\" height=\"27\" \/><\/td>\n<\/tr>\n<tr style=\"height: 29px;\">\n<td style=\"width: 675.5px; height: 29px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\n<td style=\"width: 217.828125px; height: 29px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-02.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596.\" width=\"84\" height=\"27\" \/><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"width: 675.5px; height: 24px;\">Since [latex]4[\/latex] does not go into [latex]2[\/latex], we use the first two digits of the dividend and divide [latex]25[\/latex] by [latex]4[\/latex]. The divisor [latex]4[\/latex] goes into [latex]25[\/latex] six times.<\/td>\n<td style=\"width: 217.828125px; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"width: 675.5px; height: 48px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\n<td style=\"width: 217.828125px; height: 48px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-03.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend.\" width=\"84\" height=\"46\" \/><\/td>\n<\/tr>\n<tr style=\"height: 66px;\">\n<td style=\"width: 675.5px; height: 66px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor [latex]4[\/latex] and write the product, [latex]24[\/latex], under the first two digits in the dividend.<\/td>\n<td style=\"width: 217.828125px; height: 66px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215621\/CNX_BMath_Figure_01_05_044_img-04.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend. The number 24 is written below the 25 in the dividend.\" width=\"84\" height=\"64\" \/><\/td>\n<\/tr>\n<tr style=\"height: 87px;\">\n<td style=\"width: 675.5px; height: 87px;\">Subtract that product from the first two digits in the dividend. Subtract [latex]25 - 24[\/latex] . Write the difference, [latex]1[\/latex], under the second digit in the dividend.<\/td>\n<td style=\"width: 217.828125px; height: 87px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215622\/CNX_BMath_Figure_01_05_044_img-05.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 6 is written above the 5 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24.\" width=\"84\" height=\"85\" \/><\/td>\n<\/tr>\n<tr style=\"height: 127px;\">\n<td style=\"width: 675.5px; height: 127px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]4[\/latex] fours in [latex]19[\/latex]. Write the [latex]4[\/latex] over the [latex]9[\/latex]. Multiply the [latex]4[\/latex] by [latex]4[\/latex] and subtract this product from [latex]19[\/latex].<\/td>\n<td style=\"width: 217.828125px; height: 127px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215624\/CNX_BMath_Figure_01_05_044_img-06.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 64 is written above the 59 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24. The number 9 in the dividend is written again to the right of the number 1. The number 16 is written below the number 19, and then the number 3 is written below the 6 of 16.\" width=\"84\" height=\"125\" \/><\/td>\n<\/tr>\n<tr style=\"height: 163px;\">\n<td style=\"width: 675.5px; height: 163px;\">Bring down the [latex]6[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]36[\/latex]. Write the [latex]9[\/latex] over the [latex]6[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]36[\/latex].<\/td>\n<td style=\"width: 217.828125px; height: 163px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215625\/CNX_BMath_Figure_01_05_044_img-07.png\" alt=\"The divisor 4 is placed to the left of the dividend 2596. The quotient 649 is written above the 596 in the dividend. The number 24 is written below the 25 in the dividend, and the number 1 is written below the 4 of 24. The number 9 in the dividend is written again to the right of the number 1. The number 16 is written below the number 19, and then the number 3 is written below the 6 of 16. The number 6 in the dividend is written again to the right of the number 3. The number 36 is written under the number 36. The number 0 is written below the number 6 of 36.\" width=\"84\" height=\"161\" \/><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 217.828125px; height: 12px;\"><\/td>\n<td><\/td>\n<\/tr>\n<tr style=\"height: 133px;\">\n<td style=\"width: 675.5px; height: 133px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215626\/CNX_BMath_Figure_01_05_044_img-08.png\" alt=\"The number 4 is placed below the 9 of the number 649. A multiplication symbol is placed to the far left of the number 4. the number 2596 is placed below the number 4 where the 6 of 2596 is right below the number 4.\" width=\"84\" height=\"84\" \/><\/td>\n<td style=\"width: 217.828125px; height: 133px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our arithmetic is correct.<\/p>\n<p>Add the sign of the answer from the first step:\u00a0[latex]-2,596\\div 4=-649[\/latex] .<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144636&theme=oea&iframe_resize_id=ohm144636&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]4,506\\div 6[\/latex]. Check by multiplying:<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474096\">Show Solution<\/span><\/p>\n<div id=\"q474096\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-483\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 4,506 divided by six is worked out in long division. The first line reads first we try to divide 6 into 4. Since that won't work, we try 6 into 45. There are 7 sixes in 45. We write the 7 over the 5. The expression shows the 7 above the 5 in the quotient. Multiply the 7 by the 6 and subtract the product from 45. The expression shows in long division, 45 minus 42, equals 3. The next line reads\">\n<tbody>\n<tr>\n<th style=\"width: 672.847px;\">Divide [latex]4,506\\div 6[\/latex].<\/th>\n<td style=\"width: 222.153px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215627\/CNX_BMath_Figure_01_05_045_img-01.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506.\" width=\"84\" height=\"32\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">First we try to divide [latex]6[\/latex] into [latex]4[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-02.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506.\" width=\"84\" height=\"27\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Since that won&#8217;t work, we try [latex]6[\/latex] into [latex]45[\/latex].<\/p>\n<p>There are [latex]7[\/latex] sixes in [latex]45[\/latex]. We write the [latex]7[\/latex] over the [latex]5[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-03.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 7 is placed above the 5 in the dividend.\" width=\"84\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Multiply the [latex]7[\/latex] by [latex]6[\/latex] and subtract this product from [latex]45[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215629\/CNX_BMath_Figure_01_05_045_img-04.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 7 is placed above the 5 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42.\" width=\"84\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Now bring down the [latex]0[\/latex] and repeat these steps. There are [latex]5[\/latex] sixes in [latex]30[\/latex]. Write the [latex]5[\/latex] over the [latex]0[\/latex]. Multiply the [latex]5[\/latex] by [latex]6[\/latex] and subtract this product from [latex]30[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215631\/CNX_BMath_Figure_01_05_045_img-05.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 75 is placed above the 50 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42. The 0 in the dividend is written again to the right of the number 3. Another number 30 is written below the number 30. The number 0 is written below the 0 of 30.\" width=\"84\" height=\"125\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] six in [latex]6[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply [latex]1[\/latex] by [latex]6[\/latex] and subtract this product from [latex]6[\/latex]<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215632\/CNX_BMath_Figure_01_05_045_img-06.png\" alt=\"The divisor 6 is placed to the left of the dividend 4506. The quotient 751 is placed above the 506 in the dividend. The number 42 is placed below the 45 in the dividend. The number 3 is written below the 2 of 42. The 0 in the dividend is written again to the right of the number 3. Another number 30 is written below the number 30. The number 0 is written below the 0 of 30. The 6 in the dividend is written again next to the number 0. Another number 6 is written below the number 6. The number 0 is written below the number 6.\" width=\"84\" height=\"164\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_045_img-07.png\" alt=\"The number 6 is placed below the 1 of the number 751. A multiplication symbol is placed to the far left of the number 6. The number 4506 is placed below the number 6 where the 6 in 4506 is right below the number 6.\" width=\"84\" height=\"78\" \/><\/td>\n<td style=\"width: 222.153px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144640&theme=oea&iframe_resize_id=ohm144640&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]7,263\\div -9[\/latex]. Check by multiplying.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q483321\">Show Solution<\/span><\/p>\n<div id=\"q483321\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-485\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 7,263 divided by 9 is worked out in long division. The first line states\">\n<tbody>\n<tr>\n<th style=\"width: 717.326px;\">Divide [latex]7,263\\div -9[\/latex].<\/th>\n<td style=\"width: 177.674px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Decide on the sign of the quotient.<\/td>\n<td style=\"width: 177.674px;\">\u00a0(+)\/(-) = (-)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Let&#8217;s rewrite the problem to set it up for long division using the absolute values.<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_046_img-01.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263.\" width=\"84\" height=\"30\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">First we try to divide [latex]9[\/latex] into [latex]7[\/latex].<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215634\/CNX_BMath_Figure_01_05_046_img-02.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263.\" width=\"84\" height=\"29\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Since that won&#8217;t work, we try [latex]9[\/latex] into [latex]72[\/latex]. There are [latex]8[\/latex] nines in [latex]72[\/latex]. We write the [latex]8[\/latex] over the [latex]2[\/latex].<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215635\/CNX_BMath_Figure_01_05_046_img-03.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 8 is placed above the 2 in the dividend.\" width=\"84\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Multiply the [latex]8[\/latex] by [latex]9[\/latex] and subtract this product from [latex]72[\/latex].<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215636\/CNX_BMath_Figure_01_05_046_img-04.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 8 is placed above the 2 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72.\" width=\"84\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There are [latex]0[\/latex] nines in [latex]6[\/latex]. Write the [latex]0[\/latex] over the [latex]6[\/latex]. Multiply the [latex]0[\/latex] by [latex]9[\/latex] and subtract this product from [latex]6[\/latex].<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215637\/CNX_BMath_Figure_01_05_046_img-05.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 80 is placed above the 26 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72. The 6 in the dividend is written again to the right of the number 0. Another number 0 is written below the number 6. Another number 6 is written below the number 0.\" width=\"84\" height=\"125\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]7[\/latex] nines in [latex]63[\/latex]. Write the [latex]7[\/latex] over the [latex]3[\/latex]. Multiply the [latex]7[\/latex] by [latex]9[\/latex] and subtract this product from [latex]63[\/latex].<\/td>\n<td style=\"width: 177.674px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215637\/CNX_BMath_Figure_01_05_046_img-06.png\" alt=\"The divisor 9 is placed to the left of the dividend 7263. The quotient 807 is placed above the 263 in the dividend. The number 72 is placed below the 72 in the dividend. The number 0 is written below the 2 of 72. The 6 in the dividend is written again to the right of the number 0. Another number 0 is written below the number 6. Another number 6 is written below the number 0. The 3 in the dividend is written again next to the right of the number 6. Another number 63 is written below the number 63. The number 0 is written below the 3 of the number 63.\" width=\"84\" height=\"164\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215638\/CNX_BMath_Figure_01_05_046_img-07.png\" alt=\"The number 9 is placed below the 7 of the number 751. A multiplication symbol is placed to the far left of the number 9. The number 7263 is placed below the number 9 where the 3 in 7263 is right below the number 9.\" width=\"84\" height=\"78\" \/><\/td>\n<td style=\"width: 177.674px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our arithmetic is correct.<\/p>\n<p>Add the sign of the answer from the first step:\u00a0\u00a0[latex]7,263\\div -9= -807[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Divide using long division: [latex]-764\\div 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH0022\">Show Answer<\/span><\/p>\n<div id=\"qH0022\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-191[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch this video for another example of how to use long division to divide a four-digit integer by a two-digit integer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Dividing Whole Numbers without a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/V7Korf09iWI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>So far, all the division problems have worked out evenly. For example, if we had [latex]24[\/latex] cookies and wanted to make bags of [latex]8[\/latex] cookies, we would have [latex]3[\/latex] bags. But what if there were [latex]28[\/latex] cookies and we wanted to make bags of [latex]8?[\/latex] Start with the [latex]28[\/latex] cookies.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215640\/CNX_BMath_Figure_01_05_027.png\" alt=\"28 cookies placed at random.\" width=\"316\" height=\"127\" \/><br \/>\nTry to put the cookies in groups of eight.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215642\/CNX_BMath_Figure_01_05_028.png\" alt=\"There are 28 cookies where three groups are circled with 8 cookies in each group, leaving 3 cookies outside the three circles.\" width=\"319\" height=\"141\" \/><br \/>\nThere are [latex]3[\/latex] groups of eight cookies, and [latex]4[\/latex] cookies left over. We call the [latex]4[\/latex] cookies that are left over the remainder and show it by writing R4 next to the [latex]3[\/latex]. (The R stands for <em><strong>remainder<\/strong><\/em>).<\/p>\n<p>To check this division, we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then add the remainder of [latex]4[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 3\\\\ \\hfill \\underset{\\text{___}}{\\times 8}\\\\ \\hfill 24\\\\ \\hfill \\underset{\\text{___}}{+4}\\\\ \\hfill 28\\end{array}[\/latex]<\/p>\n<p>This is an example of the <em><strong>Fundamental Theorem of Arithmetic<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Fundamental Theorem OF Arithmetic<\/h3>\n<p style=\"text-align: center;\"><span class=\"TextRun SCXW89433696 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"none\"><span class=\"NormalTextRun SCXW89433696 BCX0\">Number = Divisor \u00b7 Quotient + Remainder<\/span><\/span><span class=\"EOP SCXW89433696 BCX0\" data-ccp-props=\"{&quot;134233279&quot;:true}\">\u00a0<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]1,439\\div 4[\/latex]. Check by multiplying.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q498101\">Show Solution<\/span><\/p>\n<div id=\"q498101\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-879\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 1,439 divided by 4. The first line says to try to divide 4 into 1. Since this won't work, try 4 into 14. There are 3 fours in 14. Write the 3 over the 4. the expression shows the quotient 3 above the 4. Next, multiply the 3 by the 4 and subtract this product from 14. The expressions shows this in long division with the quotient 5 above the 3. The next line says to bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract the product from 23. The expression shows this in long division. The next line says to bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract the product from 39. There are no more numbers to bring down. There is a remainder of 3. The expression shows the answer of 359 remainder 3.\">\n<tbody>\n<tr>\n<th style=\"width: 695.851px;\">Divide [latex]1,439\\div 4[\/latex].<\/th>\n<td style=\"width: 199.149px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215643\/CNX_BMath_Figure_01_05_047_img-01.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439.\" width=\"171\" height=\"28\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">First we try to divide [latex]4[\/latex] into [latex]1[\/latex]. Since that won&#8217;t work, we try [latex]4[\/latex] into [latex]14[\/latex]. There are [latex]3[\/latex] fours in [latex]14[\/latex]. We write the [latex]3[\/latex] over the [latex]4[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215644\/CNX_BMath_Figure_01_05_047_img-02.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 3 is placed above the 4 in the dividend.\" width=\"171\" height=\"45\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Multiply the [latex]3[\/latex] by [latex]4[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215645\/CNX_BMath_Figure_01_05_047_img-03.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 3 is placed above the 4 in the dividend. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12.\" width=\"171\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]5[\/latex] fours in [latex]23[\/latex]. Write the [latex]5[\/latex] over the [latex]3[\/latex]. Multiply the [latex]5[\/latex] by [latex]4[\/latex] and subtract this product from [latex]23[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215646\/CNX_BMath_Figure_01_05_047_img-04.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 35 is placed above the 43 in the dividend. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12. The 3 in the dividend is written again to the right of the number 2. Another number 20 is written below the number 23. Another number 3 is written below the number 0in 20.\" width=\"171\" height=\"125\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]39[\/latex]. Write the [latex]9[\/latex] over the [latex]9[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]39[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]3[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215647\/CNX_BMath_Figure_01_05_047_img-05.png\" alt=\"The divisor 4 is placed to the left of the dividend 1439. The quotient 359 is placed above the 439 in the dividend. The phrase &quot;R3&quot; is placed to the right of 359. The number 12 is placed below the 14 in the dividend. The number 2 is written below the 2 of 12. The 3 in the dividend is written again to the right of the number 2. Another number 20 is written below the number 23. Another number 3 is written below the number 0in 20. The 9 in the dividend is written again next to the right of the number 3. Another number 36 is written below the number 39. The number 3 is written below the 6 of the number 36.\" width=\"171\" height=\"164\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215648\/CNX_BMath_Figure_01_05_047_img-06.png\" alt=\"The number 4 is placed below the 9 of the number 359. A multiplication symbol is placed far to the left of the number 4. The number 1436 is placed below the number 4 where the 6 in 1436 is right below the number 4. The number 3 is below the 6 in 1436. A addition symbol is placed far to the left of the number 3. The number 1439 is placed below the number 3, where 9 in 1439 is right below 6 in 1436.\" width=\"171\" height=\"125\" \/><\/td>\n<td style=\"width: 199.149px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]1,439\\div 4[\/latex] is [latex]359[\/latex] with a remainder of [latex]3[\/latex]. Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144643\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144643&theme=oea&iframe_resize_id=ohm144643&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide and then check by multiplying: [latex]-1,461\\div (-13)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q174689\">Show Solution<\/span><\/p>\n<div id=\"q174689\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-708\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\"This image shows 1,461 divided by 13. The first line says First we try to divide 13 into 1. Since that won\u2019t work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4. Multiply the 1 by 13 and subtract this product from 14. The expression shows the quotient 1 above the 4, and 13 under 14 with the difference as 1. The next line says to bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16. The expression shows the 6 in the dividend with a blue arrow pointing downward to show the 6 is brought down next to the 1. In long division, 16 minus 3 shows a difference of 3. Next it says, now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. The expression shows in long division the 1 in the dividend with a blue arrow pointing downward to show the one brought down to make 31. A 2 is placed above the 1 in the dividend. Thirty-one minus 26 is 5, but there are no more numbers to bring down, so we are done. The remainder is 5. The answer is 112 remainder 5. The next line says to check by multiplying in which the expression shows 112 with the word\">\n<tbody>\n<tr>\n<th style=\"width: 701.698px;\">[latex]-1,461\\div (-13)[\/latex].<\/th>\n<td style=\"width: 192.302px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Decide on the sign of the quotient.<\/td>\n<td style=\"width: 192.302px;\">(-)\/(-) = (+)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 192.302px;\">[latex]13\\overline{)1,461}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">First we try to divide [latex]13[\/latex] into [latex]1[\/latex]. Since that won&#8217;t work, we try [latex]13[\/latex] into [latex]14[\/latex]. There is [latex]1[\/latex] thirteen in [latex]14[\/latex]. We write the [latex]1[\/latex] over the [latex]4[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215649\/CNX_BMath_Figure_01_05_048_img-02.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 1 is placed above the 4 in the dividend.\" width=\"173\" height=\"43\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215650\/CNX_BMath_Figure_01_05_048_img-03.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 1 is placed above the 4 in the dividend. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13.\" width=\"173\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] thirteen in [latex]16[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]16[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215651\/CNX_BMath_Figure_01_05_048_img-04.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 11 is placed above the 46 in the dividend. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13. The 6 in the dividend is written again to the right of the number 1. Another number 13 is written below the number 16. Another number 3 is written below the number 3 in 13.\" width=\"173\" height=\"125\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Now bring down the [latex]1[\/latex] and repeat these steps. There are [latex]2[\/latex] thirteens in [latex]31[\/latex]. Write the [latex]2[\/latex] over the [latex]1[\/latex]. Multiply the [latex]2[\/latex] by [latex]13[\/latex] and subtract this product from [latex]31[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]5[\/latex]. [latex]1,462\\div 13[\/latex] is [latex]112[\/latex] with a remainder of [latex]5[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215652\/CNX_BMath_Figure_01_05_048_img-05.png\" alt=\"The divisor 13 is placed to the left of the dividend 1461. The quotient 112 is placed above the 461 in the dividend. The phrase &quot;R5&quot; is placed to the right of 112. The number 13 is placed below the 14 in the dividend. The number 1 is written below the 3 of 13. The 6 in the dividend is written again to the right of the number 1. Another number 13 is written below the number 16. Another number 3 is written below the number 3 in 13. The 1 in the dividend is written again next to the right of the number 3. Another number 26 is written below the number 31. The number 5 is written below the 6 of the number 26.\" width=\"173\" height=\"164\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215653\/CNX_BMath_Figure_01_05_048_img-06.png\" alt=\"The number 13 (divisor) is placed below the 12 of the number 112 (quotient). A multiplication symbol is placed far to the left of the number 13. The number 336 is placed below the number 13 where the 6 in 336 is right below the number 3 in 13. The number 1120 is placed below the number 336 where 0 in 1120 is right below the 6 in 336. Another number 5 (remainder) is placed below the number 1120. A addition symbol is placed far to the left of the number 5. The number 1461 is placed below the number 5, where 1 in 1461 is right below the number 5.\" width=\"173\" height=\"136\" \/><\/td>\n<td style=\"width: 192.302px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our arithmetic is correct.<\/p>\n<p>Add the sign of the answer from the first step:\u00a0[latex]-1,461\\div (-13)=+(112\\text{R}5)= 112\\text{R}5[\/latex] Notice that when the quotient is positive, so is the remainder.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144644\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144644&theme=oea&iframe_resize_id=ohm144644&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide and check by multiplying: [latex]-74,521\\div 241[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862075\">Show Solution<\/span><\/p>\n<div id=\"q862075\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-787\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 74,521 divided by 241. First we try to divide 241 into 7. Since that won\u2019t work, we try 241 into 74. That still won\u2019t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3. Since 3 times 241 equals 723, we write the 3 over the 5 in 745. The image shows the expression in long division with the quotient 3 above the 5. Next it says to bring down the 2 and repeat these steps. 241 does not divide into 222. Write a 0 over the 2 as a place holder and then continue. Note that 4 would be too large because 4 times 241 equals 964, which is greater than 745. Multiply the 3 by 241 and subtract this product from 745. The image shows the expression in long division 745 minus 723, which is 22. Next, it says to bring down the 2 and repeat these steps. 241 does not divide into 222. Write a 0 over the 2 as a place holder and then continue. Next, it says to bring down the 1 and repeat these steps. Try 9 Since 9 times 241 equals 2,169, we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221. The expression shows the one brought down and the subtraction of 2,221 minus 2,169 which gives the answer as 309 remainder 52. The image is checked through multiplication. The numbers 309 times 241are multiplied vertically. The expression is worked out to show that 74,3521 checks and the answer is correct.\">\n<tbody>\n<tr>\n<th style=\"width: 692.424px;\">[latex]-74,521\\div 241[\/latex].<\/th>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Decide on the sign of the quotient.<\/td>\n<td style=\"width: 202.576px;\">(-)\/(+) = (-)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 202.576px;\">[latex]241\\overline{)74,521}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">First we try to divide [latex]241[\/latex] into [latex]7[\/latex]. Since that won\u2019t work, we try [latex]241[\/latex] into [latex]74[\/latex]. That still won\u2019t work, so we try [latex]241[\/latex] into[latex]745[\/latex]. Since [latex]2[\/latex] divides into [latex]7[\/latex] three times, we try [latex]3[\/latex]. Since [latex]3\\times 241=723[\/latex] , we write the [latex]3[\/latex] over the [latex]5[\/latex] in [latex]745[\/latex]. Note that [latex]4[\/latex] would be too large, because [latex]4\\times 241=964[\/latex] , which is greater than [latex]745[\/latex].<\/td>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Multiply the [latex]3[\/latex] by [latex]241[\/latex] and subtract this product from [latex]745[\/latex].<\/td>\n<td style=\"width: 202.576px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215654\/CNX_BMath_Figure_01_05_049_img-02.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 3 is placed above the 5 in the dividend. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723.\" width=\"175\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Now bring down the [latex]2[\/latex] and repeat these steps. [latex]241[\/latex] does not divide into [latex]222[\/latex].<\/p>\n<p>We write a [latex]0[\/latex] over the [latex]2[\/latex] as a placeholder and then continue.<\/td>\n<td style=\"width: 202.576px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215655\/CNX_BMath_Figure_01_05_049_img-03.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 30 is placed above the 52 in the dividend. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723. The 2 in the dividend is written again to the right of the number 22.\" width=\"175\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Now bring down the [latex]1[\/latex] and repeat these steps. Try [latex]9[\/latex]. Since [latex]9\\times 241=2,169[\/latex] , we write the [latex]9[\/latex] over the [latex]1[\/latex]. Multiply the [latex]9[\/latex] by [latex]241[\/latex] and subtract this product from [latex]2,221[\/latex].<\/td>\n<td style=\"width: 202.576px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215656\/CNX_BMath_Figure_01_05_049_img-04.png\" alt=\"The divisor 241 is placed to the left of the dividend 74521. The quotient 309 is placed above the 521 in the dividend. The phrase &quot;R52&quot; is placed to the right of 309. The number 723 is placed below the 745 in the dividend. The number 22 is written below the 23 of 723. The 21 in the dividend is written again to the right of the number 22. Another number 2169 is written below the number 2221. Another number 52 is written below 69 in 2169.\" width=\"175\" height=\"125\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">There are no more numbers to bring down, so we are finished. The remainder is [latex]52[\/latex]. So [latex]74,521\\div 241[\/latex] is [latex]309[\/latex] with a remainder of [latex]52[\/latex].<\/td>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 692.424px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215657\/CNX_BMath_Figure_01_05_049_img-05.png\" alt=\"The number 241 (divisor) is placed below the number 309 (quotient). A multiplication symbol is placed far to the left of the number 241. The number 309 is placed below the number 241. The number 12360 is placed below the number 309. The number 61800 is placed below the number 12360. Another number 74469 is placed below the number 61800. Another number 52 (remainder) is placed below the number 72469. A addition symbol is placed far to the left of the number 52. The number 74521 is placed below the number 52.\" width=\"167\" height=\"198\" \/><\/td>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our arithmetic is correct.<\/p>\n<p>Add the sign of the answer from the first step:\u00a0 [latex]-74,521\\div 241= -\\left(309\\text{R}52\\right)[\/latex]. When the quotient is negative, so is the remainder, so this simplifies to\u00a0[latex]-74,521\\div 241= -309\\text{R}(-52)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Divide 8452 by \u20139.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q910292\">Show Answer<\/span><\/p>\n<div id=\"q910292\" class=\"hidden-answer\" style=\"display: none\">\u2013939 R(1)<\/div>\n<\/div>\n<\/div>\n<p>Watch the video below for another example of how to use long division to divide integers when there is a remainder.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Example:  Dividing Whole Numbers with a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UPUcShGCBOs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2><\/h2>\n","protected":false},"author":422605,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1114","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1114","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":26,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1114\/revisions"}],"predecessor-version":[{"id":3220,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1114\/revisions\/3220"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1114\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1114"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1114"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1114"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1114"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}