{"id":34,"date":"2023-06-21T13:22:26","date_gmt":"2023-06-21T13:22:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/prime-factorization-and-least-common-multiple\/"},"modified":"2023-08-07T01:28:57","modified_gmt":"2023-08-07T01:28:57","slug":"prime-factorization-and-least-common-multiple","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/prime-factorization-and-least-common-multiple\/","title":{"raw":"RP1.1   Prime Factorization and the Least Common Multiple","rendered":"RP1.1   Prime Factorization and the Least Common Multiple"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the prime factorization of a number<\/li>\r\n \t<li>Find the least common multiple of a list of numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe word <em>factor<\/em> can be both a noun and a verb. To factor a number is to rewrite it by breaking it up into a product of smaller numbers. For example, we can factor 24 by writing it as [latex]6\\ast4[\/latex]. We say that 6 and 4 are factors of 24. But so are 2 and 12. There are many ways to write this number.\r\n\r\nA useful skill to have when doing algebra is the ability to rewrite numbers and expressions in helpful forms.\u00a0 For example, consider some different forms of the number 24.\r\n<p style=\"text-align: center;\">[latex]24 \\qquad \\dfrac{72}{3} \\qquad \\sqrt{576} \\qquad 6\\ast4 \\qquad 2\\cdot2\\cdot2\\cdot3 [\/latex].<\/p>\r\n<strong>Composite numbers<\/strong>, like 24, are natural numbers that can be written as products of other natural numbers. <strong>Prime numbers<\/strong>\u00a0are natural numbers that have only two possible factors, themselves and the number 1.\u00a0The final form of 24 in the list above, [latex]2\\cdot2\\cdot2\\cdot3 [\/latex], is called its\u00a0<strong>prime factorization<\/strong>. When we write the prime factorization of a number, we are writing it as a product of only its prime factors. Being able to find the prime factorization of a composite number is an especially useful skill to have when doing algebra.\r\n<h3 class=\"title\">Prime Factorization<\/h3>\r\nThe prime factorization of a number is the product of prime numbers that equals the number.\r\n\r\nYou may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.\r\n\r\n[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]\r\n\r\n<strong>Tip<\/strong>: Knowing the first five prime numbers will come in handy when reducing fractions.\r\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\r\nOne way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a \"branch\" of the factor tree.\r\n\r\nIf a factor is prime, we circle it (like a bud on a tree), and do not factor that \"branch\" any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.\r\n\r\nWe continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.\r\n\r\nFor example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12[\/latex]. We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" \/>\r\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" \/>\r\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" \/>\r\nThe prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.\r\n<p style=\"text-align: center;\">[latex]2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\r\nIn cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\r\nNote that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the result would have been the same if we had started with [latex]2[\/latex] and [latex]18, 4[\/latex] and [latex]9,\\text{ or }6\\text{ and }6[\/latex].\r\n<h3 class=\"title\">Find the prime factorization of a composite number using the tree method<\/h3>\r\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\r\n \t<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\r\n \t<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\r\n \t<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\r\n \t<li>Write the composite number as the product of all the circled primes.<\/li>\r\n<\/ol>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the prime factorization of [latex]48[\/latex] using the factor tree method.\r\n\r\nSolution:\r\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\r\n<tbody>\r\n<tr>\r\n<td>We can start our tree using any factor pair of [latex]48[\/latex]. Let's use [latex]2\\text{ and }24[\/latex].\r\nWe circle the [latex]2[\/latex] because it is prime and so that branch is complete.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now we will factor [latex]24[\/latex]. Let's use [latex]4\\text{ and }6[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Neither factor is prime, so we do not circle either.We factor the [latex]4[\/latex], using [latex]2\\text{ and }2[\/latex].We factor [latex]6\\text{, using }2\\text{ and }3[\/latex].\r\nWe circle the [latex]2\\text{s and the }3[\/latex] since they are prime. Now all of the branches end in a prime.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the product of the circled numbers.<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in exponential form.<\/td>\r\n<td>[latex]{2}^{4}\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCheck this on your own by multiplying all the factors together. The result should be [latex]48[\/latex].\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]146554[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows how to find the prime factorization of [latex]60[\/latex] using the factor tree method.\r\n\r\nhttps:\/\/youtu.be\/2K5pBvb7Sss\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the prime factorization of [latex]84[\/latex] using the factor tree method.\r\n[reveal-answer q=\"214088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214088\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467446629\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows two factor trees with the number 84 at the top. In the first tree two branches are splitting out from under 84. The branches use the factor pair 4 and 21 with 21 at the end of the right branch and 4 at the end of the left branch. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 21. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 21 use the factor pair 3 and 7. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 84 is made up of all of the circled numbers from the factor tree which is 2, 2, 3, and 7. The prime factorization can be written as 2 times 2 times 3 times 7 or using exponents for repeated multiplication of 2 it can be written as 2 squared times 3 times 7.\">\r\n<tbody>\r\n<tr>\r\n<td>We start with the factor pair [latex]4\\text{ and }21[\/latex].\r\nNeither factor is prime so we factor them further.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220027\/CNX_BMath_Figure_02_05_023_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now the factors are all prime, so we circle them.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220029\/CNX_BMath_Figure_02_05_023_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then we write [latex]84[\/latex] as the product of all circled primes.<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 3\\cdot 7[\/latex][latex]{2}^{2}\\cdot 3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDraw a factor tree of [latex]84[\/latex].\r\n\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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145453&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nThere are other methods that work well to find the prime factorization of a number. Any method that factors out small primes repeatedly until there are only prime factors remaining is acceptable. See the video below for a demonstration of using stacked division to find a prime factorization.\r\n\r\nhttps:\/\/youtu.be\/V_wBWdndCuw\r\n<h3>Finding Least Common Multiples<\/h3>\r\nConsider the multiples of two numbers, 3 and 4. What is the least common multiple between them?\r\n\r\nCertainly 3 and 4 have many multiples in common: 12, 24, 36, etc. But 12 is the smallest among them.\r\n\r\nThe smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the LCM of [latex]3[\/latex] and [latex]4[\/latex] is [latex]12[\/latex].\r\n\r\nOne of the reasons we find prime factorizations\u00a0is to use them\u00a0to find the least common multiple of two or more numbers. This will be useful when we add and subtract fractions with different denominators.\r\n<h3>Prime Factors Method<\/h3>\r\nWe can find the least common multiple of two numbers by inspecting 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 3 \\qquad[\/latex]\u00a0 and [latex] \\qquad 18=2\\cdot 3\\cdot 3[\/latex]<\/p>\r\nThen we find the largest instance of each prime appearing in any one factorization. Here, we see that the number 2 appears twice in the factorization of 12, but only once in 18. And the number 3 appears twice in 18, but only once in 12. So, we select [latex]2\\cdot2[\/latex] and\u00a0 [latex]3\\cdot3[\/latex] and multiply them together to find the LCM.\r\n<p style=\"padding-left: 30px;\">[latex]12=2\\cdot 2\\cdot 3 \\qquad \\text{ and } \\qquad 18=2\\cdot 3\\cdot 3[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]\\qquad 2\\cdot2 \\qquad \\qquad \\ast \\qquad \\qquad 3\\cdot3 [\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]\\qquad\\quad \\qquad 2\\cdot2\\cdot3\\cdot3=36[\/latex].<\/p>\r\nThe LCM of 12 and 18 is 36.\r\n\r\nIf a prime appears the same number of times in any factorization, just select one instance. Let's find the LCM of a list of numbers as an example.\r\n\r\nFind the LCM of 12, 18, 20, and 60. First we'll write the numbers as products of primes.\r\n<p style=\"padding-left: 30px;\">[latex]12 = 2\\cdot2\\cdot3 \\qquad 18=2\\cdot3\\cdot3 \\qquad 20=2\\cdot2\\cdot5 \\qquad 60=2\\cdot2\\cdot3\\cdot5[\/latex]<\/p>\r\nWe see that the largest instance of the number 2 appears in the numbers 12, 20, and 60. But we just select one largest instance. The largest instance of the number 3 appears in 18, where there are two factors of 3. And the number 5 appears once in 20 and once in 60, so we'll select one 5.\r\n\r\nThe LCM of 12, 18, 20, and 60 is\r\n<p style=\"padding-left: 30px;\">[latex]2\\cdot2\\cdot3\\cdot3\\cdot5 = 180[\/latex].<\/p>\r\nSo, 180 is the smallest number that 12, 18, 20, and 60 all divide evenly into.\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>Inspect each factorization for the largest number of instances of each prime number appearing in each factorization<\/li>\r\n \t<li>Select one set of each largest instance of each prime factor appearing.<\/li>\r\n \t<li>Multiply the selected factors together to obtain the LCM.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIt can be helpful to write the prime factors next to each number, matching them vertically in columns, then bringing down the primes in each column to collect the factors whose product is the LCM. The following examples illustrate this technique.\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 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\/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]The 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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145459&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\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][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]The 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 see how to find the Least Common Multiple by using prime factorization.\r\n\r\nhttps:\/\/youtu.be\/hZvRDG-HgMY","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the prime factorization of a number<\/li>\n<li>Find the least common multiple of a list of numbers<\/li>\n<\/ul>\n<\/div>\n<p>The word <em>factor<\/em> can be both a noun and a verb. To factor a number is to rewrite it by breaking it up into a product of smaller numbers. For example, we can factor 24 by writing it as [latex]6\\ast4[\/latex]. We say that 6 and 4 are factors of 24. But so are 2 and 12. There are many ways to write this number.<\/p>\n<p>A useful skill to have when doing algebra is the ability to rewrite numbers and expressions in helpful forms.\u00a0 For example, consider some different forms of the number 24.<\/p>\n<p style=\"text-align: center;\">[latex]24 \\qquad \\dfrac{72}{3} \\qquad \\sqrt{576} \\qquad 6\\ast4 \\qquad 2\\cdot2\\cdot2\\cdot3[\/latex].<\/p>\n<p><strong>Composite numbers<\/strong>, like 24, are natural numbers that can be written as products of other natural numbers. <strong>Prime numbers<\/strong>\u00a0are natural numbers that have only two possible factors, themselves and the number 1.\u00a0The final form of 24 in the list above, [latex]2\\cdot2\\cdot2\\cdot3[\/latex], is called its\u00a0<strong>prime factorization<\/strong>. When we write the prime factorization of a number, we are writing it as a product of only its prime factors. Being able to find the prime factorization of a composite number is an especially useful skill to have when doing algebra.<\/p>\n<h3 class=\"title\">Prime Factorization<\/h3>\n<p>The prime factorization of a number is the product of prime numbers that equals the number.<\/p>\n<p>You may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.<\/p>\n<p>[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]<\/p>\n<p><strong>Tip<\/strong>: Knowing the first five prime numbers will come in handy when reducing fractions.<\/p>\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\n<p>One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a &#8220;branch&#8221; of the factor tree.<\/p>\n<p>If a factor is prime, we circle it (like a bud on a tree), and do not factor that &#8220;branch&#8221; any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.<\/p>\n<p>We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.<\/p>\n<p>For example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12[\/latex]. We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.<\/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\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" \/><br \/>\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].<\/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\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" \/><br \/>\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].<\/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\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" \/><br \/>\nThe prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\n<p>Note that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the result would have been the same if we had started with [latex]2[\/latex] and [latex]18, 4[\/latex] and [latex]9,\\text{ or }6\\text{ and }6[\/latex].<\/p>\n<h3 class=\"title\">Find the prime factorization of a composite number using the tree method<\/h3>\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\n<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\n<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\n<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\n<li>Write the composite number as the product of all the circled primes.<\/li>\n<\/ol>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the prime factorization of [latex]48[\/latex] using the factor tree method.<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\n<tbody>\n<tr>\n<td>We can start our tree using any factor pair of [latex]48[\/latex]. Let&#8217;s use [latex]2\\text{ and }24[\/latex].<br \/>\nWe circle the [latex]2[\/latex] because it is prime and so that branch is complete.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now we will factor [latex]24[\/latex]. Let&#8217;s use [latex]4\\text{ and }6[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Neither factor is prime, so we do not circle either.We factor the [latex]4[\/latex], using [latex]2\\text{ and }2[\/latex].We factor [latex]6\\text{, using }2\\text{ and }3[\/latex].<br \/>\nWe circle the [latex]2\\text{s and the }3[\/latex] since they are prime. Now all of the branches end in a prime.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the product of the circled numbers.<\/td>\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write in exponential form.<\/td>\n<td>[latex]{2}^{4}\\cdot 3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Check this on your own by multiplying all the factors together. The result should be [latex]48[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146554\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146554&theme=oea&iframe_resize_id=ohm146554&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows how to find the prime factorization of [latex]60[\/latex] using the factor tree method.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Prime Factorization\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2K5pBvb7Sss?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the prime factorization of [latex]84[\/latex] using the factor tree method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q214088\">Show Solution<\/span><\/p>\n<div id=\"q214088\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467446629\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The figure shows two factor trees with the number 84 at the top. In the first tree two branches are splitting out from under 84. The branches use the factor pair 4 and 21 with 21 at the end of the right branch and 4 at the end of the left branch. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 21. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 21 use the factor pair 3 and 7. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 84 is made up of all of the circled numbers from the factor tree which is 2, 2, 3, and 7. The prime factorization can be written as 2 times 2 times 3 times 7 or using exponents for repeated multiplication of 2 it can be written as 2 squared times 3 times 7.\">\n<tbody>\n<tr>\n<td>We start with the factor pair [latex]4\\text{ and }21[\/latex].<br \/>\nNeither factor is prime so we factor them further.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220027\/CNX_BMath_Figure_02_05_023_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now the factors are all prime, so we circle them.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220029\/CNX_BMath_Figure_02_05_023_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then we write [latex]84[\/latex] as the product of all circled primes.<\/td>\n<td>[latex]2\\cdot 2\\cdot 3\\cdot 7[\/latex][latex]{2}^{2}\\cdot 3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Draw a factor tree of [latex]84[\/latex].<\/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=\"ohm145453\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145453&#38;theme=oea&#38;iframe_resize_id=ohm145453&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>There are other methods that work well to find the prime factorization of a number. Any method that factors out small primes repeatedly until there are only prime factors remaining is acceptable. See the video below for a demonstration of using stacked division to find a prime factorization.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1: Prime Factorization Using Stacked Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/V_wBWdndCuw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Finding Least Common Multiples<\/h3>\n<p>Consider the multiples of two numbers, 3 and 4. What is the least common multiple between them?<\/p>\n<p>Certainly 3 and 4 have many multiples in common: 12, 24, 36, etc. But 12 is the smallest among them.<\/p>\n<p>The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the LCM of [latex]3[\/latex] and [latex]4[\/latex] is [latex]12[\/latex].<\/p>\n<p>One of the reasons we find prime factorizations\u00a0is to use them\u00a0to find the least common multiple of two or more numbers. This will be useful when we add and subtract fractions with different denominators.<\/p>\n<h3>Prime Factors Method<\/h3>\n<p>We can find the least common multiple of two numbers by inspecting 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 3 \\qquad[\/latex]\u00a0 and [latex]\\qquad 18=2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>Then we find the largest instance of each prime appearing in any one factorization. Here, we see that the number 2 appears twice in the factorization of 12, but only once in 18. And the number 3 appears twice in 18, but only once in 12. So, we select [latex]2\\cdot2[\/latex] and\u00a0 [latex]3\\cdot3[\/latex] and multiply them together to find the LCM.<\/p>\n<p style=\"padding-left: 30px;\">[latex]12=2\\cdot 2\\cdot 3 \\qquad \\text{ and } \\qquad 18=2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\qquad 2\\cdot2 \\qquad \\qquad \\ast \\qquad \\qquad 3\\cdot3[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\qquad\\quad \\qquad 2\\cdot2\\cdot3\\cdot3=36[\/latex].<\/p>\n<p>The LCM of 12 and 18 is 36.<\/p>\n<p>If a prime appears the same number of times in any factorization, just select one instance. Let&#8217;s find the LCM of a list of numbers as an example.<\/p>\n<p>Find the LCM of 12, 18, 20, and 60. First we&#8217;ll write the numbers as products of primes.<\/p>\n<p style=\"padding-left: 30px;\">[latex]12 = 2\\cdot2\\cdot3 \\qquad 18=2\\cdot3\\cdot3 \\qquad 20=2\\cdot2\\cdot5 \\qquad 60=2\\cdot2\\cdot3\\cdot5[\/latex]<\/p>\n<p>We see that the largest instance of the number 2 appears in the numbers 12, 20, and 60. But we just select one largest instance. The largest instance of the number 3 appears in 18, where there are two factors of 3. And the number 5 appears once in 20 and once in 60, so we&#8217;ll select one 5.<\/p>\n<p>The LCM of 12, 18, 20, and 60 is<\/p>\n<p style=\"padding-left: 30px;\">[latex]2\\cdot2\\cdot3\\cdot3\\cdot5 = 180[\/latex].<\/p>\n<p>So, 180 is the smallest number that 12, 18, 20, and 60 all divide evenly into.<\/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>Inspect each factorization for the largest number of instances of each prime number appearing in each factorization<\/li>\n<li>Select one set of each largest instance of each prime factor appearing.<\/li>\n<li>Multiply the selected factors together to obtain the LCM.<\/li>\n<\/ol>\n<\/div>\n<p>It can be helpful to write the prime factors next to each number, matching them vertically in columns, then bringing down the primes in each column to collect the factors whose product is the LCM. The following examples illustrate this technique.<\/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 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\/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]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=\"ohm145459\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145459&#38;theme=oea&#38;iframe_resize_id=ohm145459&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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][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]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 see how to find the Least Common Multiple by using prime factorization.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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\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-34\">\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>Ex 1: Prime Factorization. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2K5pBvb7Sss\">https:\/\/youtu.be\/2K5pBvb7Sss<\/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 1: Prime Factorization Using Stacked Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/V_wBWdndCuw\">https:\/\/youtu.be\/V_wBWdndCuw<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 146554, 145453, 145459, 145462. <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":395986,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"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\":\"original\",\"description\":\"Ex 1: Prime Factorization\",\"author\":\"James Sousa 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