{"id":1389,"date":"2021-11-01T16:58:26","date_gmt":"2021-11-01T16:58:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1389"},"modified":"2022-09-01T18:40:23","modified_gmt":"2022-09-01T18:40:23","slug":"1-3-2-multiples-and-the-least-common-multiple","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-3-2-multiples-and-the-least-common-multiple\/","title":{"raw":"1.3.2: Multiples and the Least Common Multiple","rendered":"1.3.2: Multiples and the Least Common Multiple"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Identify multiples of whole numbers<\/li>\r\n \t<li>Find the least common multiple by listing multiples<\/li>\r\n \t<li>Find the least common multiple of two numbers by prime factorization<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Multiple<\/strong>: the product of a whole number and a natural number<\/li>\r\n \t<li><strong>Common multiple<\/strong>:\u00a0a number that is a multiple of two or more numbers<\/li>\r\n \t<li><strong>Least common multiple<\/strong>: the smallest number\u00a0that is a multiple of two or more numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Identifying Multiples of a Number<\/h2>\r\nAnnie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn\u2019t have to count each one. She counts by twos: [latex]2,4,6,8,10,12[\/latex]. She has [latex]12[\/latex] shoes in her closet.\r\n\r\nThe numbers [latex]2,4,6,8,10,12[\/latex] are called <em><strong>multiples<\/strong><\/em> of [latex]2[\/latex]. Multiples of [latex]2[\/latex] can be written as the product of a natural number and [latex]2[\/latex]. The first six multiples of [latex]2[\/latex] are given below.\r\n<p style=\"text-align: left; padding-left: 30px;\">[latex]\\begin{array}{l}1\\cdot 2=2\\\\ 2\\cdot 2=4\\\\ 3\\cdot 2=6\\\\ 4\\cdot 2=8\\\\ 5\\cdot 2=10\\\\ 6\\cdot 2=12\\end{array}[\/latex]<\/p>\r\nA multiple of a number is the product of the number and a counting number. So a multiple of [latex]3[\/latex] would be the product of a counting number and [latex]3[\/latex]. Below are the first six multiples of [latex]3[\/latex].\r\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}1\\cdot 3=3\\\\ 2\\cdot 3=6\\\\ 3\\cdot 3=9\\\\ 4\\cdot 3=12\\\\ 5\\cdot 3=15\\\\ 6\\cdot 3=18\\end{array}[\/latex]<\/p>\r\nWe can find the multiples of any number by continuing this process. The table below\u00a0shows the multiples of [latex]2[\/latex] through [latex]9[\/latex] for the first twelve counting numbers.\r\n<table id=\"fs-id1946698\" summary=\"The table has nine rows and thirteen columns. The first row is a header row. The first row, first column labels the first row as the counting numbers. The remaining columns are labeled 1 through 12. Starting in the second row, the first column list the categories: \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Counting Number<\/strong><\/th>\r\n<th>[latex]1[\/latex]<\/th>\r\n<th>[latex]2[\/latex]<\/th>\r\n<th>[latex]3[\/latex]<\/th>\r\n<th>[latex]4[\/latex]<\/th>\r\n<th>[latex]5[\/latex]<\/th>\r\n<th>[latex]6[\/latex]<\/th>\r\n<th>[latex]7[\/latex]<\/th>\r\n<th>[latex]8[\/latex]<\/th>\r\n<th>[latex]9[\/latex]<\/th>\r\n<th>[latex]10[\/latex]<\/th>\r\n<th>[latex]11[\/latex]<\/th>\r\n<th>[latex]12[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]22[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<td>[latex]21[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]27[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]33[\/latex]<\/td>\r\n<td>[latex]36[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]28[\/latex]<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<td>[latex]36[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]45[\/latex]<\/td>\r\n<td>[latex]50[\/latex]<\/td>\r\n<td>[latex]55[\/latex]<\/td>\r\n<td>[latex]60[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }6[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]36[\/latex]<\/td>\r\n<td>[latex]42[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<td>[latex]54[\/latex]<\/td>\r\n<td>[latex]60[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }7[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]21[\/latex]<\/td>\r\n<td>[latex]28[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]42[\/latex]<\/td>\r\n<td>[latex]49[\/latex]<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]70[\/latex]<\/td>\r\n<td>[latex]77[\/latex]<\/td>\r\n<td>[latex]84[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }8[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<td>[latex]80[\/latex]<\/td>\r\n<td>[latex]88[\/latex]<\/td>\r\n<td>[latex]96[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{Multiples of }9[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<td>[latex]27[\/latex]<\/td>\r\n<td>[latex]36[\/latex]<\/td>\r\n<td>[latex]45[\/latex]<\/td>\r\n<td>[latex]54[\/latex]<\/td>\r\n<td>[latex]63[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<td>[latex]90[\/latex]<\/td>\r\n<td>[latex]99[\/latex]<\/td>\r\n<td>[latex]108[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiple of a Number<\/h3>\r\nA number is a multiple of [latex]n[\/latex] if it is the product of a natural number and [latex]n[\/latex].\r\n\r\n<\/div>\r\nRecognizing the patterns for multiples will be helpful to you as you continue in this course.\r\n<h2>Finding the Least Common Multiple of Two Numbers<\/h2>\r\nOne of the reasons we find\u00a0multiples and primes is to use them\u00a0to find the <em><strong>least common multiple<\/strong><\/em> of two numbers. This will be useful when we add and subtract fractions with different denominators.\r\n<h3>Listing Multiples Method<\/h3>\r\nA common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of [latex]10[\/latex] and [latex]25[\/latex]. We can list the first several multiples of each number. Then we look for multiples that are common to both lists\u2014these are the common multiples.\r\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{c}10\\text{ : }10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110\\ldots \\hfill \\\\ 25\\text{ : }25, 50,75, 100, 125\\ldots \\hfill \\end{array}[\/latex]<\/p>\r\nWe see that [latex]50[\/latex] and [latex]100[\/latex] appear in both lists. They are common multiples of [latex]10[\/latex] and [latex]25[\/latex]. We would find more common multiples if we continued the list of multiples for each.\r\n\r\nThe smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of [latex]10[\/latex] and [latex]25[\/latex] is [latex]50[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Find the least common multiple (LCM) of two numbers by listing multiples<\/h3>\r\n<ol id=\"eip-id1168466318223\" class=\"stepwise\">\r\n \t<li>List the first several multiples of each number.<\/li>\r\n \t<li>Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.<\/li>\r\n \t<li>Look for the smallest number that is common to both lists.<\/li>\r\n \t<li>This number is the LCM.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the LCM of [latex]15[\/latex] and [latex]20[\/latex] by listing multiples.\r\n\r\nSolution:\r\nList the first several multiples of [latex]15[\/latex] and of [latex]20[\/latex]. Identify the first common multiple.\r\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}\\text{15: }15,30,45,60,75,90,105,120\\hfill \\\\ \\text{20: }20,40,60,80,100,120,140,160\\hfill \\end{array}[\/latex]<\/p>\r\nThe smallest number to appear on both lists is [latex]60[\/latex], so [latex]60[\/latex] is the least common multiple of [latex]15[\/latex] and [latex]20[\/latex].\r\n\r\nNotice that [latex]120[\/latex] is on both lists, too. It is a common multiple, but it is not the least common multiple.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145458[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show an example of how to find the Least Common Multiple by listing multiples of each number.\r\n\r\nhttps:\/\/youtu.be\/7twRSmgcrLM\r\n<h3>Prime Factors Method<\/h3>\r\nAnother way to find the least common multiple of two numbers is to use their prime factors. We\u2019ll use this method to find the LCM of [latex]12[\/latex] and [latex]18[\/latex].\r\n\r\nWe start by finding the prime factorization of each number.\r\n<p style=\"padding-left: 30px;\">[latex]12=2\\cdot 2\\cdot 318=2\\cdot 3\\cdot 3[\/latex]<\/p>\r\nThen we write each number as a product of primes, matching primes vertically when possible.\r\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}12=2\\cdot 2\\cdot 3\\hfill \\\\ 18=2\\cdot 3\\cdot 3\\end{array}[\/latex]<\/p>\r\nNow we bring down a single prime from each column. The LCM is the product of these primes.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220036\/CNX_BMath_Figure_02_05_006_img.png\" alt=\"The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.\" \/>\r\nNotice that the prime factors of [latex]12[\/latex] and the prime factors of [latex]18[\/latex] are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that [latex]36[\/latex] is the least common multiple.\r\n<div class=\"textbox shaded\">\r\n<h3>Find the LCM using the prime factors method<\/h3>\r\n<ol id=\"eip-id1168469871370\" class=\"stepwise\">\r\n \t<li>Find the prime factorization of each number.<\/li>\r\n \t<li>Write each number as a product of primes, matching primes vertically when possible.<\/li>\r\n \t<li>Bring down a single prime from each column.<\/li>\r\n \t<li>Multiply the factors to get the LCM.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the LCM of [latex]15[\/latex] and [latex]18[\/latex] using the prime factors method.\r\n[reveal-answer q=\"628602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"628602\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467200222\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"In the figure the prime factorization of 15 is written as the equation 15 equals 3 times 5. Next to that is the prime factorization of 18 written as the equation 2 times 3 times 3. Below that is the prime factorization equations of 15 and 18 written to align vertically with the equation for 15 above the equation for 18. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 3 times 3 times 5. Arrows are drawn down vertically from the prime factorization of 15 through the prime factorization of 18 ending at the LCM equation. Since there is no 2 in the prime factorization of 15, the first arrow starts at the 2 in the prime factorization of 18 and points down to the 2 in the LCM. The second arrow starts at the first 3 in the prime factorization of 15 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. Since there are no more three's in the prime factorization of 15, the next arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM. Since there are no more factors in the prime factorization of 18, the last arrow starts at the 5 in the prime factorization of 15 and points down to through the empty space at the end of the prime factorization of 18 to the 5 in the LCM. The least common multiple of 15 and 18 is 2 times 3 times 3 times 5 which is 90.\">\r\n<tbody>\r\n<tr>\r\n<td>Write each number as a product of primes.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220037\/CNX_BMath_Figure_02_05_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write each number as a product of primes, matching primes vertically when possible.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220038\/CNX_BMath_Figure_02_05_026_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Bring down a single primes from each column.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220040\/CNX_BMath_Figure_02_05_026_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the factors to get the LCM.<\/td>\r\n<td>[latex]\\text{LCM}=2\\cdot 3\\cdot 3\\cdot 5[\/latex]\r\n\r\nThe LCM of [latex]15\\text{ and }18\\text{ is } 90[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145459&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"230\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the LCM of [latex]50[\/latex] and [latex]100[\/latex] using the prime factors method.\r\n[reveal-answer q=\"500769\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500769\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466277374\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"In the figure the prime factorization of 50 is written as the equation 50 equals 2 times 5 times 5. Next to that is the prime factorization of 100 written as the equation 2 times 2 times 5 times 5. Below that is the prime factorization equations of 50 and 100 written to align vertically with the equation for 50 above the equation for 100. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 100. Below this line is the equation LCM equal to 2 times 2 times 5 times 5. Arrows are drawn down vertically from the prime factorization of 50 through the prime factorization of 100 ending at the LCM equation. Since there is only one 2 in the prime factorization of 50, the first arrow starts at the first 2 in the prime factorization of 100 and points down to the first 2 in the LCM. The second arrow starts at the first 2 in the prime factorization of 50 and continues down through the second 2 in the prime factorization of 100. Ending with the second 2 in the LCM. The next arrow starts at the first 5 in the prime factorization of 50 and continues through the first 5 in the prime factorization of 100. Ending in the first 5 in the LCM. The last arrow starts at the second 5 in the prime factorization of 50 and continues through the second 5 in the prime factorization of 100. Ending in the second 5 in the LCM. The least common multiple of 50 and 100 is 2 times 2 times 5 times 5 which is 100.\">\r\n<tbody>\r\n<tr>\r\n<td>Write the prime factorization of each number.<\/td>\r\n<td>[latex]50=2\\cdot{5}\\cdot{5}\\quad\\quad\\quad{100=2\\cdot{2}\\cdot{5}\\cdot{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write each number as a product of primes, matching primes vertically when possible.<\/td>\r\n<td>[latex]50=\\quad{2\\cdot{5}\\cdot{5}}[\/latex]\r\n\r\n[latex]100=2\\cdot{2}\\cdot{5}\\cdot{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Bring down the primes in each column.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220044\/CNX_BMath_Figure_02_05_027_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the factors to get the LCM.<\/td>\r\n<td>[latex]\\text{LCM}=2\\cdot 2\\cdot 5\\cdot 5[\/latex]\r\n\r\nThe LCM of [latex]50\\text{ and } 100\\text{ is } 100[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145462[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show how to find the Least Common Multiple by using prime factorization.\r\n\r\nhttps:\/\/youtu.be\/hZvRDG-HgMY\r\n<h2>Finding the Least Common Multiple of More Than Two Numbers<\/h2>\r\nThe prime factorization method can be applied to more than two numbers.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the least common multiple of [latex]12, 20, \\text{and }35.[\/latex]\r\n\r\n<strong>Solution<\/strong>\r\n\r\nFirst write the prime factorization of each number:\r\n\r\n[latex]12=2\\cdot 2\\cdot 3[\/latex]\r\n\r\n[latex]20=2\\cdot 2\\cdot 5[\/latex]\r\n\r\n[latex]35=5\\cdot 7[\/latex]\r\n\r\nNow line them up in matching columns, bring down a single prime from each, and multiply the prime factors:\r\n\r\n<img class=\"aligncenter size-medium wp-image-1573\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/09232020\/LCM-300x208.png\" alt=\"LCM\" width=\"300\" height=\"208\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nFind the least common multiple of [latex]42, 60, 14,\\text{and }36[\/latex].\r\n\r\n<strong>Solution<\/strong>\r\n\r\nFirst write the prime factorization of each number:\r\n\r\n[latex]42=2\\cdot 3\\cdot 7[\/latex]\r\n\r\n[latex]60=2\\cdot 2\\cdot 3\\cdot 5[\/latex]\r\n\r\n[latex]14=2\\cdot 7[\/latex]\r\n\r\n[latex]36=2\\cdot 2\\cdot 3\\cdot 3[\/latex]\r\n\r\nNow line them up in matching columns, bring down a single prime from each, and multiply the prime factors:\r\n\r\n<img class=\"aligncenter size-medium wp-image-1575\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/09233426\/LCM2-300x189.png\" alt=\"LCM example\" width=\"300\" height=\"189\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the least common multiple of [latex]14, 22, 48[\/latex].\r\n\r\n[reveal-answer q=\"H1261\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H1261\"]\r\n\r\n[latex]14=2\\cdot 7[\/latex]\r\n\r\n[latex]22=2\\cdot 11[\/latex]\r\n\r\n[latex]48=2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]\r\n\r\n[latex]\\text{LCM }=3696[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Identify multiples of whole numbers<\/li>\n<li>Find the least common multiple by listing multiples<\/li>\n<li>Find the least common multiple of two numbers by prime factorization<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Multiple<\/strong>: the product of a whole number and a natural number<\/li>\n<li><strong>Common multiple<\/strong>:\u00a0a number that is a multiple of two or more numbers<\/li>\n<li><strong>Least common multiple<\/strong>: the smallest number\u00a0that is a multiple of two or more numbers<\/li>\n<\/ul>\n<\/div>\n<h2>Identifying Multiples of a Number<\/h2>\n<p>Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn\u2019t have to count each one. She counts by twos: [latex]2,4,6,8,10,12[\/latex]. She has [latex]12[\/latex] shoes in her closet.<\/p>\n<p>The numbers [latex]2,4,6,8,10,12[\/latex] are called <em><strong>multiples<\/strong><\/em> of [latex]2[\/latex]. Multiples of [latex]2[\/latex] can be written as the product of a natural number and [latex]2[\/latex]. The first six multiples of [latex]2[\/latex] are given below.<\/p>\n<p style=\"text-align: left; padding-left: 30px;\">[latex]\\begin{array}{l}1\\cdot 2=2\\\\ 2\\cdot 2=4\\\\ 3\\cdot 2=6\\\\ 4\\cdot 2=8\\\\ 5\\cdot 2=10\\\\ 6\\cdot 2=12\\end{array}[\/latex]<\/p>\n<p>A multiple of a number is the product of the number and a counting number. So a multiple of [latex]3[\/latex] would be the product of a counting number and [latex]3[\/latex]. Below are the first six multiples of [latex]3[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}1\\cdot 3=3\\\\ 2\\cdot 3=6\\\\ 3\\cdot 3=9\\\\ 4\\cdot 3=12\\\\ 5\\cdot 3=15\\\\ 6\\cdot 3=18\\end{array}[\/latex]<\/p>\n<p>We can find the multiples of any number by continuing this process. The table below\u00a0shows the multiples of [latex]2[\/latex] through [latex]9[\/latex] for the first twelve counting numbers.<\/p>\n<table id=\"fs-id1946698\" summary=\"The table has nine rows and thirteen columns. The first row is a header row. The first row, first column labels the first row as the counting numbers. The remaining columns are labeled 1 through 12. Starting in the second row, the first column list the categories:\">\n<thead>\n<tr valign=\"top\">\n<th><strong>Counting Number<\/strong><\/th>\n<th>[latex]1[\/latex]<\/th>\n<th>[latex]2[\/latex]<\/th>\n<th>[latex]3[\/latex]<\/th>\n<th>[latex]4[\/latex]<\/th>\n<th>[latex]5[\/latex]<\/th>\n<th>[latex]6[\/latex]<\/th>\n<th>[latex]7[\/latex]<\/th>\n<th>[latex]8[\/latex]<\/th>\n<th>[latex]9[\/latex]<\/th>\n<th>[latex]10[\/latex]<\/th>\n<th>[latex]11[\/latex]<\/th>\n<th>[latex]12[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]22[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<td>[latex]21[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]27[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]33[\/latex]<\/td>\n<td>[latex]36[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]28[\/latex]<\/td>\n<td>[latex]32[\/latex]<\/td>\n<td>[latex]36[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]44[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]25[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]45[\/latex]<\/td>\n<td>[latex]50[\/latex]<\/td>\n<td>[latex]55[\/latex]<\/td>\n<td>[latex]60[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }6[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]36[\/latex]<\/td>\n<td>[latex]42[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<td>[latex]54[\/latex]<\/td>\n<td>[latex]60[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }7[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]21[\/latex]<\/td>\n<td>[latex]28[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]42[\/latex]<\/td>\n<td>[latex]49[\/latex]<\/td>\n<td>[latex]56[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]77[\/latex]<\/td>\n<td>[latex]84[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }8[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]32[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<td>[latex]56[\/latex]<\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<td>[latex]80[\/latex]<\/td>\n<td>[latex]88[\/latex]<\/td>\n<td>[latex]96[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{Multiples of }9[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<td>[latex]27[\/latex]<\/td>\n<td>[latex]36[\/latex]<\/td>\n<td>[latex]45[\/latex]<\/td>\n<td>[latex]54[\/latex]<\/td>\n<td>[latex]63[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<td>[latex]81[\/latex]<\/td>\n<td>[latex]90[\/latex]<\/td>\n<td>[latex]99[\/latex]<\/td>\n<td>[latex]108[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<h3>Multiple of a Number<\/h3>\n<p>A number is a multiple of [latex]n[\/latex] if it is the product of a natural number and [latex]n[\/latex].<\/p>\n<\/div>\n<p>Recognizing the patterns for multiples will be helpful to you as you continue in this course.<\/p>\n<h2>Finding the Least Common Multiple of Two Numbers<\/h2>\n<p>One of the reasons we find\u00a0multiples and primes is to use them\u00a0to find the <em><strong>least common multiple<\/strong><\/em> of two numbers. This will be useful when we add and subtract fractions with different denominators.<\/p>\n<h3>Listing Multiples Method<\/h3>\n<p>A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of [latex]10[\/latex] and [latex]25[\/latex]. We can list the first several multiples of each number. Then we look for multiples that are common to both lists\u2014these are the common multiples.<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{c}10\\text{ : }10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110\\ldots \\hfill \\\\ 25\\text{ : }25, 50,75, 100, 125\\ldots \\hfill \\end{array}[\/latex]<\/p>\n<p>We see that [latex]50[\/latex] and [latex]100[\/latex] appear in both lists. They are common multiples of [latex]10[\/latex] and [latex]25[\/latex]. We would find more common multiples if we continued the list of multiples for each.<\/p>\n<p>The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of [latex]10[\/latex] and [latex]25[\/latex] is [latex]50[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the least common multiple (LCM) of two numbers by listing multiples<\/h3>\n<ol id=\"eip-id1168466318223\" class=\"stepwise\">\n<li>List the first several multiples of each number.<\/li>\n<li>Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.<\/li>\n<li>Look for the smallest number that is common to both lists.<\/li>\n<li>This number is the LCM.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the LCM of [latex]15[\/latex] and [latex]20[\/latex] by listing multiples.<\/p>\n<p>Solution:<br \/>\nList the first several multiples of [latex]15[\/latex] and of [latex]20[\/latex]. Identify the first common multiple.<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}\\text{15: }15,30,45,60,75,90,105,120\\hfill \\\\ \\text{20: }20,40,60,80,100,120,140,160\\hfill \\end{array}[\/latex]<\/p>\n<p>The smallest number to appear on both lists is [latex]60[\/latex], so [latex]60[\/latex] is the least common multiple of [latex]15[\/latex] and [latex]20[\/latex].<\/p>\n<p>Notice that [latex]120[\/latex] is on both lists, too. It is a common multiple, but it is not the least common multiple.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145458\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145458&theme=oea&iframe_resize_id=ohm145458&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show an example of how to find the Least Common Multiple by listing multiples of each number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Determining the Least Common Multiple Using a List of Multiples\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7twRSmgcrLM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Prime Factors Method<\/h3>\n<p>Another way to find the least common multiple of two numbers is to use their prime factors. We\u2019ll use this method to find the LCM of [latex]12[\/latex] and [latex]18[\/latex].<\/p>\n<p>We start by finding the prime factorization of each number.<\/p>\n<p style=\"padding-left: 30px;\">[latex]12=2\\cdot 2\\cdot 318=2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>Then we write each number as a product of primes, matching primes vertically when possible.<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\begin{array}{l}12=2\\cdot 2\\cdot 3\\hfill \\\\ 18=2\\cdot 3\\cdot 3\\end{array}[\/latex]<\/p>\n<p>Now we bring down a single prime from each column. The LCM is the product of these primes.<\/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\/24220036\/CNX_BMath_Figure_02_05_006_img.png\" alt=\"The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.\" \/><br \/>\nNotice that the prime factors of [latex]12[\/latex] and the prime factors of [latex]18[\/latex] are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that [latex]36[\/latex] is the least common multiple.<\/p>\n<div class=\"textbox shaded\">\n<h3>Find the LCM using the prime factors method<\/h3>\n<ol id=\"eip-id1168469871370\" class=\"stepwise\">\n<li>Find the prime factorization of each number.<\/li>\n<li>Write each number as a product of primes, matching primes vertically when possible.<\/li>\n<li>Bring down a single prime from each column.<\/li>\n<li>Multiply the factors to get the LCM.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the LCM of [latex]15[\/latex] and [latex]18[\/latex] using the prime factors method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q628602\">Show Solution<\/span><\/p>\n<div id=\"q628602\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467200222\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"In the figure the prime factorization of 15 is written as the equation 15 equals 3 times 5. Next to that is the prime factorization of 18 written as the equation 2 times 3 times 3. Below that is the prime factorization equations of 15 and 18 written to align vertically with the equation for 15 above the equation for 18. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 3 times 3 times 5. Arrows are drawn down vertically from the prime factorization of 15 through the prime factorization of 18 ending at the LCM equation. Since there is no 2 in the prime factorization of 15, the first arrow starts at the 2 in the prime factorization of 18 and points down to the 2 in the LCM. The second arrow starts at the first 3 in the prime factorization of 15 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. Since there are no more three's in the prime factorization of 15, the next arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM. Since there are no more factors in the prime factorization of 18, the last arrow starts at the 5 in the prime factorization of 15 and points down to through the empty space at the end of the prime factorization of 18 to the 5 in the LCM. The least common multiple of 15 and 18 is 2 times 3 times 3 times 5 which is 90.\">\n<tbody>\n<tr>\n<td>Write each number as a product of primes.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220037\/CNX_BMath_Figure_02_05_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write each number as a product of primes, matching primes vertically when possible.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220038\/CNX_BMath_Figure_02_05_026_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Bring down a single primes from each column.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220040\/CNX_BMath_Figure_02_05_026_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the factors to get the LCM.<\/td>\n<td>[latex]\\text{LCM}=2\\cdot 3\\cdot 3\\cdot 5[\/latex]<\/p>\n<p>The LCM of [latex]15\\text{ and }18\\text{ is } 90[\/latex].<\/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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145459&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"230\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the LCM of [latex]50[\/latex] and [latex]100[\/latex] using the prime factors method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q500769\">Show Solution<\/span><\/p>\n<div id=\"q500769\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466277374\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"In the figure the prime factorization of 50 is written as the equation 50 equals 2 times 5 times 5. Next to that is the prime factorization of 100 written as the equation 2 times 2 times 5 times 5. Below that is the prime factorization equations of 50 and 100 written to align vertically with the equation for 50 above the equation for 100. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 100. Below this line is the equation LCM equal to 2 times 2 times 5 times 5. Arrows are drawn down vertically from the prime factorization of 50 through the prime factorization of 100 ending at the LCM equation. Since there is only one 2 in the prime factorization of 50, the first arrow starts at the first 2 in the prime factorization of 100 and points down to the first 2 in the LCM. The second arrow starts at the first 2 in the prime factorization of 50 and continues down through the second 2 in the prime factorization of 100. Ending with the second 2 in the LCM. The next arrow starts at the first 5 in the prime factorization of 50 and continues through the first 5 in the prime factorization of 100. Ending in the first 5 in the LCM. The last arrow starts at the second 5 in the prime factorization of 50 and continues through the second 5 in the prime factorization of 100. Ending in the second 5 in the LCM. The least common multiple of 50 and 100 is 2 times 2 times 5 times 5 which is 100.\">\n<tbody>\n<tr>\n<td>Write the prime factorization of each number.<\/td>\n<td>[latex]50=2\\cdot{5}\\cdot{5}\\quad\\quad\\quad{100=2\\cdot{2}\\cdot{5}\\cdot{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write each number as a product of primes, matching primes vertically when possible.<\/td>\n<td>[latex]50=\\quad{2\\cdot{5}\\cdot{5}}[\/latex]<\/p>\n<p>[latex]100=2\\cdot{2}\\cdot{5}\\cdot{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Bring down the primes in each column.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220044\/CNX_BMath_Figure_02_05_027_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the factors to get the LCM.<\/td>\n<td>[latex]\\text{LCM}=2\\cdot 2\\cdot 5\\cdot 5[\/latex]<\/p>\n<p>The LCM of [latex]50\\text{ and } 100\\text{ is } 100[\/latex].<\/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=\"ohm145462\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145462&theme=oea&iframe_resize_id=ohm145462&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show how to find the Least Common Multiple by using prime factorization.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Determining the Least Common Multiple Using Prime Factorization\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hZvRDG-HgMY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding the Least Common Multiple of More Than Two Numbers<\/h2>\n<p>The prime factorization method can be applied to more than two numbers.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the least common multiple of [latex]12, 20, \\text{and }35.[\/latex]<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First write the prime factorization of each number:<\/p>\n<p>[latex]12=2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>[latex]20=2\\cdot 2\\cdot 5[\/latex]<\/p>\n<p>[latex]35=5\\cdot 7[\/latex]<\/p>\n<p>Now line them up in matching columns, bring down a single prime from each, and multiply the prime factors:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1573\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/09232020\/LCM-300x208.png\" alt=\"LCM\" width=\"300\" height=\"208\" \/><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Find the least common multiple of [latex]42, 60, 14,\\text{and }36[\/latex].<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First write the prime factorization of each number:<\/p>\n<p>[latex]42=2\\cdot 3\\cdot 7[\/latex]<\/p>\n<p>[latex]60=2\\cdot 2\\cdot 3\\cdot 5[\/latex]<\/p>\n<p>[latex]14=2\\cdot 7[\/latex]<\/p>\n<p>[latex]36=2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>Now line them up in matching columns, bring down a single prime from each, and multiply the prime factors:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1575\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/09233426\/LCM2-300x189.png\" alt=\"LCM example\" width=\"300\" height=\"189\" \/><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the least common multiple of [latex]14, 22, 48[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH1261\">Show Answer<\/span><\/p>\n<div id=\"qH1261\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]14=2\\cdot 7[\/latex]<\/p>\n<p>[latex]22=2\\cdot 11[\/latex]<\/p>\n<p>[latex]48=2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>[latex]\\text{LCM }=3696[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1389\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Finding the Least Common Multiple of More Than Two Numbers. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Determine Multiples of a Given Number.. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning.. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mkEWqspRVKk\">https:\/\/youtu.be\/mkEWqspRVKk<\/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: Determining the Least Common Multiple Using a List of Multiples. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/7twRSmgcrLM\">https:\/\/youtu.be\/7twRSmgcrLM<\/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: Determining the Least Common Multiple Using Prime Factorization. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hZvRDG-HgMY\">https:\/\/youtu.be\/hZvRDG-HgMY<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":422605,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Determine Multiples of a Given Number.\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning.\",\"organization\":\"\",\"url\":\" https:\/\/youtu.be\/mkEWqspRVKk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example: Determining the Least Common Multiple Using a List of Multiples\",\"author\":\"James Sousa 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