{"id":4604,"date":"2020-04-21T00:19:07","date_gmt":"2020-04-21T00:19:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/dividing-whole-numbers-properties-of-division\/"},"modified":"2024-06-26T17:17:38","modified_gmt":"2024-06-26T17:17:38","slug":"dividing-whole-numbers-properties-of-division","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/dividing-whole-numbers-properties-of-division\/","title":{"raw":"Dividing Whole Numbers: Properties of Division","rendered":"Dividing Whole Numbers: Properties of Division"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Divide whole numbers and check the answer using multiplication<\/li>\n \t<li>Identify and apply the division properties of one<\/li>\n \t<li>Identify and apply the division properties of zero<\/li>\n \t<li>Use the long division algorithm to divide multiple-digit numbers<\/li>\n \t<li>Identify the divisor, dividend, and remainder in a division problem<\/li>\n<\/ul>\n<\/div>\n<h2>Divide Whole Numbers<\/h2>\nWe said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of 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.\n\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].\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide. Then check by multiplying.\n<ol>\n \t<li>[latex]42\\div 6[\/latex]<\/li>\n \t<li>[latex]\\frac{72}{9}[\/latex]<\/li>\n \t<li>[latex]7\\overline{)63}[\/latex]<\/li>\n<\/ol>\nSolution:\n<table id=\"eip-id1168287031935\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\n<tbody>\n<tr>\n<td>1.<\/td>\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.\n\n[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<td>2.<\/td>\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.\n\n[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<td>3.<\/td>\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.\n\n[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&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144463[\/ohm_question]\n\n<\/div>\n&nbsp;\n\nWhat is the quotient when you divide a number by itself?\n<p style=\"text-align: center;\">[latex]\\frac{15}{15}=1[\/latex]<\/p>\n<p style=\"text-align: center;\">because [latex]1\\cdot 15=15[\/latex]<\/p>\nDividing any number [latex]\\text{(except 0)}[\/latex] by itself produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.\n<div class=\"textbox shaded\">\n<h3>Division Properties of One<\/h3>\n<table id=\"eip-735\" summary=\"a\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14.4585px;\">\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide. Then check by multiplying:\n<ol id=\"eip-id1168288568257\" class=\"circled\">\n \t<li>[latex]11\\div 11[\/latex]<\/li>\n \t<li>[latex]\\frac{19}{1}[\/latex]<\/li>\n \t<li>[latex]1\\overline{)7}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"519474\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"519474\"]\n\nSolution:\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]11\\div 11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.\n\n[latex]1\\cdot 11[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11\\quad\\checkmark [\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.\n\n[latex]19\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]19\\quad\\checkmark [\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n<table id=\"eip-id1168289599719\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\overline{)7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.\n\n[latex]7\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]7\\quad\\checkmark [\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144635[\/ohm_question]\n\n<\/div>\n&nbsp;\n\nSuppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].\n\nNow suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex] to get [latex]10[\/latex]. This cannot happen because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <em>undefined<\/em>.\n\nThese two ideas make up the Division Properties of Zero.\n<div class=\"textbox shaded\">\n<h3>Division Properties of Zero<\/h3>\n<table id=\"eip-158\" summary=\"a\">\n<tbody>\n<tr>\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\n<td>[latex]0\\div a=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by zero is undefined.<\/td>\n<td>[latex]a\\div 0[\/latex] undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n&nbsp;\n\nAnother way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away [latex]0[\/latex] from [latex]10?[\/latex] Because subtracting [latex]0[\/latex] will never change the total, we will never get an answer. So we cannot divide a number by [latex]0[\/latex].\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide. Check by multiplying:\n<ol>\n \t<li>[latex]0\\div 3[\/latex]<\/li>\n \t<li>[latex]\\frac{10}{0}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"208505\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"208505\"]\n\nSolution\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Zero divided by any number is zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.\n\n[latex]0\\cdot 3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0\\quad\\checkmark [\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10\/0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Division by zero is undefined.<\/td>\n<td>undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144478[\/ohm_question]\n\n<\/div>\n&nbsp;\n\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 long division. Let\u2019s work through the process by dividing [latex]78[\/latex] by [latex]3[\/latex].\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<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 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=\"CNX_BMath_Figure_01_05_043_img-02.png\"><\/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 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=\"CNX_BMath_Figure_01_05_043_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_043_img-04.png\"><\/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 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=\"CNX_BMath_Figure_01_05_043_img-05.png\"><\/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 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=\"CNX_BMath_Figure_01_05_043_img-06.png\"><\/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 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=\"CNX_BMath_Figure_01_05_043_img-07.png\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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.\n<p style=\"text-align: center;\">[latex]\\text{So }78\\div 3=26[\/latex].<\/p>\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].\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; padding-left: 60px;\">It does, so our answer is correct.&nbsp;[latex]\\checkmark[\/latex]<\/p>\n\n<div class=\"textbox shaded\">\n<h3>Divide whole numbers<\/h3>\n<ol id=\"eip-id1168288534169\" class=\"stepwise\">\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>\n \t<li>Write the quotient above the dividend.<\/li>\n \t<li>Multiply the quotient by the divisor and write the product under the dividend.<\/li>\n \t<li>Subtract that product from the dividend.<\/li>\n \t<li>Bring down the next digit of the dividend.<\/li>\n \t<li>Repeat from Step 1 until there are no more digits in the dividend to bring down.<\/li>\n \t<li>Check by multiplying the quotient times the divisor.<\/li>\n<\/ol>\n<\/div>\nIn the video below we show another example of using long division.\n\nhttps:\/\/youtu.be\/KvVhaB5mqr8\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and then check by multiplying:\n\n[latex]2,596\\div 4[\/latex]\n[reveal-answer q=\"252445\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"252445\"]\n\nSolution\n<table id=\"eip-287\" class=\"unnumbered unstyled\" style=\"width: 917px;\" 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<td style=\"width: 676.753px;\">Let's rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-01.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-02.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-03.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-04.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-05.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-06.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img 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=\"CNX_BMath_Figure_01_05_044_img-07.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">So [latex]2,596\\div 4=649[\/latex] .<\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Check by multiplying.\n\n<img 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=\"CNX_BMath_Figure_01_05_044_img-08.png\"><\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIt equals the dividend, so our answer is correct.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144636[\/ohm_question]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and then check by multiplying:\n\n[latex]4,506\\div 6[\/latex]\n[reveal-answer q=\"474096\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"474096\"]\n\nSolution\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<td style=\"width: 672.847px;\">Let's rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 222.153px;\"><img 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=\"CNX_BMath_Figure_01_05_045_img-01.png\"><\/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 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=\"CNX_BMath_Figure_01_05_045_img-02.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Since that won't work, we try [latex]6[\/latex] into [latex]45[\/latex].\n\nThere 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 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=\"CNX_BMath_Figure_01_05_045_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_045_img-04.png\"><\/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 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=\"CNX_BMath_Figure_01_05_045_img-05.png\"><\/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 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=\"CNX_BMath_Figure_01_05_045_img-06.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Check by multiplying.\n\n<img 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=\"CNX_BMath_Figure_01_05_045_img-07.png\"><\/td>\n<td style=\"width: 222.153px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIt equals the dividend, so our answer is correct.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144640[\/ohm_question]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and then check by multiplying:\n\n[latex]7,263\\div 9[\/latex]\n[reveal-answer q=\"483321\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"483321\"]\n\nSolution\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<td style=\"width: 717.326px;\">Let's rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 177.674px;\"><img 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=\"CNX_BMath_Figure_01_05_046_img-01.png\"><\/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 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=\"CNX_BMath_Figure_01_05_046_img-02.png\"><\/td>\n<\/tr>\n<tr>\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>\n<td style=\"width: 177.674px;\"><img 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=\"CNX_BMath_Figure_01_05_046_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_046_img-04.png\"><\/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 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=\"CNX_BMath_Figure_01_05_046_img-05.png\"><\/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 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=\"CNX_BMath_Figure_01_05_046_img-06.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Check by multiplying.\n\n<img 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=\"CNX_BMath_Figure_01_05_046_img-07.png\"><\/td>\n<td style=\"width: 177.674px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIt equals the dividend, so our answer is correct.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144640[\/ohm_question]\n\n<\/div>\nWatch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.\n\nhttps:\/\/youtu.be\/V7Korf09iWI\n\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.\n\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=\"An image of 28 cookies placed at random.\">\nTry to put the cookies in groups of eight.\n\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=\"An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.\">\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 remainder.)\n\nTo check this division we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then <strong>add the remainder<\/strong> of [latex]4[\/latex].\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\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and then check by multiplying:\n\n[latex]1,439\\div 4[\/latex]\n[reveal-answer q=\"498101\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"498101\"]\n\nSolution\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<td style=\"width: 695.851px;\">Let's rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 199.149px;\"><img 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=\"CNX_BMath_Figure_01_05_047_img-01.png\"><\/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'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 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=\"CNX_BMath_Figure_01_05_047_img-02.png\"><\/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 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=\"CNX_BMath_Figure_01_05_047_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_047_img-04.png\"><\/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 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=\"CNX_BMath_Figure_01_05_047_img-05.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Check by multiplying.\n\n<img 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=\"CNX_BMath_Figure_01_05_047_img-06.png\"><\/td>\n<td style=\"width: 199.149px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSo [latex]1,439\\div 4[\/latex] is [latex]359[\/latex] with a remainder of [latex]3[\/latex]. Our answer is correct.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144643[\/ohm_question]\n\n<\/div>\n&nbsp;\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and then check by multiplying:\n\n[latex]1,461\\div 13[\/latex]\n[reveal-answer q=\"174689\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"174689\"]\n\nSolution\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<td style=\"width: 701.698px;\">Let'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'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 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=\"CNX_BMath_Figure_01_05_048_img-02.png\"><\/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 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=\"CNX_BMath_Figure_01_05_048_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_048_img-04.png\"><\/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 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=\"CNX_BMath_Figure_01_05_048_img-05.png\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Check by multiplying.\n\n<img 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=\"CNX_BMath_Figure_01_05_048_img-06.png\"><\/td>\n<td style=\"width: 192.302px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nOur answer is correct.\n\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144644[\/ohm_question]\n\n<\/div>\n&nbsp;\n\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.\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\nDivide and check by multiplying:\n\n[latex]74,521\\div 241[\/latex]\n[reveal-answer q=\"862075\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"862075\"]\n\nSolution\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<td style=\"width: 692.424px;\">Let'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 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=\"CNX_BMath_Figure_01_05_049_img-02.png\"><\/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].\n\nWe write a [latex]0[\/latex] over the [latex]2[\/latex] as a placeholder and then continue.<\/td>\n<td style=\"width: 202.576px;\"><img 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=\"CNX_BMath_Figure_01_05_049_img-03.png\"><\/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 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=\"CNX_BMath_Figure_01_05_049_img-04.png\"><\/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.\n\n<img 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=\"CNX_BMath_Figure_01_05_049_img-05.png\"><\/td>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n&nbsp;\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n[ohm_question]144645[\/ohm_question]\n\n<\/div>\nWatch the video below for another example of how to use long division to divide whole numbers when there is a remainder.\n\nhttps:\/\/youtu.be\/UPUcShGCBOs\n\n&nbsp;\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide whole numbers and check the answer using multiplication<\/li>\n<li>Identify and apply the division properties of one<\/li>\n<li>Identify and apply the division properties of zero<\/li>\n<li>Use the long division algorithm to divide multiple-digit numbers<\/li>\n<li>Identify the divisor, dividend, and remainder in a division problem<\/li>\n<\/ul>\n<\/div>\n<h2>Divide Whole Numbers<\/h2>\n<p>We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of 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<td>1.<\/td>\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<td>2.<\/td>\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<td>3.<\/td>\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<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144463\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144463&theme=oea&iframe_resize_id=ohm144463&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>What is the quotient when you divide a number by itself?<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{15}{15}=1[\/latex]<\/p>\n<p style=\"text-align: center;\">because [latex]1\\cdot 15=15[\/latex]<\/p>\n<p>Dividing any number [latex]\\text{(except 0)}[\/latex] by itself produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of One<\/h3>\n<table id=\"eip-735\" summary=\"a\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14.4585px;\">\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Then check by multiplying:<\/p>\n<ol id=\"eip-id1168288568257\" class=\"circled\">\n<li>[latex]11\\div 11[\/latex]<\/li>\n<li>[latex]\\frac{19}{1}[\/latex]<\/li>\n<li>[latex]1\\overline{)7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q519474\">Show Solution<\/span><\/p>\n<div id=\"q519474\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]11\\div 11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]1\\cdot 11[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]19\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]19\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168289599719\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\overline{)7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]7\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]7\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144635&theme=oea&iframe_resize_id=ohm144635&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Suppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].<\/p>\n<p>Now suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex] to get [latex]10[\/latex]. This cannot happen because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <em>undefined<\/em>.<\/p>\n<p>These two ideas make up the Division Properties of Zero.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of Zero<\/h3>\n<table id=\"eip-158\" summary=\"a\">\n<tbody>\n<tr>\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\n<td>[latex]0\\div a=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by zero is undefined.<\/td>\n<td>[latex]a\\div 0[\/latex] undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away [latex]0[\/latex] from [latex]10?[\/latex] Because subtracting [latex]0[\/latex] will never change the total, we will never get an answer. So we cannot divide a number by [latex]0[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Check by multiplying:<\/p>\n<ol>\n<li>[latex]0\\div 3[\/latex]<\/li>\n<li>[latex]\\frac{10}{0}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q208505\">Show Solution<\/span><\/p>\n<div id=\"q208505\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Zero divided by any number is zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]0\\cdot 3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10\/0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Division by zero is undefined.<\/td>\n<td>undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144478\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144478&theme=oea&iframe_resize_id=ohm144478&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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 long division. 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<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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-05.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-06.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_043_img-07.png\" \/><\/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; padding-left: 60px;\">It does, so our answer is correct.&nbsp;[latex]\\checkmark[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Divide whole numbers<\/h3>\n<ol id=\"eip-id1168288534169\" class=\"stepwise\">\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<\/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 and then check by multiplying:<\/p>\n<p>[latex]2,596\\div 4[\/latex]<\/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;\" 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<td style=\"width: 676.753px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-01.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-02.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-03.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-04.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-05.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-06.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">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: 218.247px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-07.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">So [latex]2,596\\div 4=649[\/latex] .<\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Check by multiplying.<\/p>\n<p><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_044_img-08.png\" \/><\/td>\n<td style=\"width: 218.247px;\"><\/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<p>&nbsp;<\/p>\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 and then check by multiplying:<\/p>\n<p>[latex]4,506\\div 6[\/latex]<\/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<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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-01.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-05.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-06.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Check by multiplying.<\/p>\n<p><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_045_img-07.png\" \/><\/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<p>&nbsp;<\/p>\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 and then check by multiplying:<\/p>\n<p>[latex]7,263\\div 9[\/latex]<\/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<td style=\"width: 717.326px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 177.674px;\"><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-01.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-05.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-06.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 717.326px;\">Check by multiplying.<\/p>\n<p><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_046_img-07.png\" \/><\/td>\n<td style=\"width: 177.674px;\"><\/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<p>&nbsp;<\/p>\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>Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.<\/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 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=\"An image of 28 cookies placed at random.\" \/><br \/>\nTry to put the cookies in groups of eight.<\/p>\n<p><img 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=\"An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.\" \/><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 remainder.)<\/p>\n<p>To check this division we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then <strong>add the remainder<\/strong> 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<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide and then check by multiplying:<\/p>\n<p>[latex]1,439\\div 4[\/latex]<\/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<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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-01.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-05.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Check by multiplying.<\/p>\n<p><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_047_img-06.png\" \/><\/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<p>&nbsp;<\/p>\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:<\/p>\n<p>[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<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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_048_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_048_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_048_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_048_img-05.png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Check by multiplying.<\/p>\n<p><img decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_048_img-06.png\" \/><\/td>\n<td style=\"width: 192.302px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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<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 exercises\">\n<h3>example<\/h3>\n<p>Divide and check by multiplying:<\/p>\n<p>[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<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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_049_img-02.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_049_img-03.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_049_img-04.png\" \/><\/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 decoding=\"async\" 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=\"CNX_BMath_Figure_01_05_049_img-05.png\" \/><\/td>\n<td style=\"width: 202.576px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144645\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144645&theme=oea&iframe_resize_id=ohm144645&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the video below for another example of how to use long division to divide whole numbers 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\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4604\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Example: Dividing Whole Numbers without a Remainder. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/V7Korf09iWI\">https:\/\/youtu.be\/V7Korf09iWI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Long Division - Two Digit Divided by One Digit (No Remainder). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KvVhaB5mqr8\">https:\/\/youtu.be\/KvVhaB5mqr8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Example: Dividing Whole Numbers with a Remainder. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UPUcShGCBOs\">https:\/\/youtu.be\/UPUcShGCBOs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 144463, 143012, 143012, 144636, 144640, 144643, 144644, 144645. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Example: Dividing Whole Numbers without a Remainder\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/V7Korf09iWI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Long Division - Two Digit Divided by One Digit (No Remainder)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KvVhaB5mqr8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example: Dividing Whole Numbers with a Remainder\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/UPUcShGCBOs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Question ID: 144463, 143012, 143012, 144636, 144640, 144643, 144644, 144645\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4604","chapter","type-chapter","status-publish","hentry"],"part":4588,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4604","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4604\/revisions"}],"predecessor-version":[{"id":5385,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4604\/revisions\/5385"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/4588"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4604\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4604"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4604"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4604"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}