{"id":9444,"date":"2017-05-02T22:33:46","date_gmt":"2017-05-02T22:33:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9444"},"modified":"2024-04-29T18:48:20","modified_gmt":"2024-04-29T18:48:20","slug":"finding-the-prime-factorization-of-a-composite-number","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-the-prime-factorization-of-a-composite-number\/","title":{"raw":"Finding the Prime Factorization of a Composite Number","rendered":"Finding the Prime Factorization of a Composite Number"},"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 using the factor tree method<\/li>\r\n \t<li>Find the prime factorization of a number using the ladder method<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the previous section, we found the factors of a number. Prime numbers have only two factors, the number [latex]1[\/latex] and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.\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 same 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 class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end.\" width=\"293\" height=\"108\" \/><\/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 class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end. Two more branches are splitting out from under 24. The left branch has the number 4 at the end and the right branch has the number 6 at the end.\" width=\"293\" height=\"181\" \/><\/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].\r\n\r\nWe 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 class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end. Two more branches are splitting out from under 24. The left branch has the number 4 at the end and the right branch has the number 6 at the end. 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. Two more branches are splitting out from under 6. The left branch has the number 2 at the end with a circle around it, and the right branch has the number 3 at the end with a circle around it. \" width=\"293\" height=\"266\" \/><\/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 class=\"alignnone\" 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=\"The figure shows a factor tree with the number 84 at the top. Two branches are splitting out from under 84. The left branch has a number 4 at the end. The right branch has the number 21 at the end.\" width=\"311\" height=\"104\" \/><\/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 class=\"alignnone\" 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=\"The figure shows a factor tree with the number 84 at the top. Two branches are splitting out from under 84. The left branch has a number 4 at the end. The right branch has the number 21 at the end. Two more branches are splitting out from under 4. The both the left and right branch has the number 2 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. Two more branches are splitting out from under 21. The left branch has the number 3 at the end with a circle around it, and the right branch has the number 7 at the end with a circle around it. \" width=\"311\" height=\"196\" \/><\/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]\r\n\r\n[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<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145453&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Prime Factorization Using the Ladder Method<\/h2>\r\nThe ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.\r\n\r\nTo begin building the \"ladder,\" divide the given number by its smallest prime factor. For example, to start the ladder for [latex]36[\/latex], we divide [latex]36[\/latex] by [latex]2[\/latex], the smallest prime factor of [latex]36[\/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\/24220030\/CNX_BMath_Figure_02_05_010_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket.\" \/>\r\nTo add a \"step\" to the ladder, we continue dividing by the same prime until it no longer divides evenly.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220031\/CNX_BMath_Figure_02_05_011_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket.\" \/>\r\nThen we divide by the next prime; so we divide [latex]9[\/latex] by [latex]3[\/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\/24220031\/CNX_BMath_Figure_02_05_012_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket. Another division bracket is written around the 9 with a 3 on the outside left of the bracket and a 3 above the 9, outside of the bracket.\" \/>\r\nWe continue dividing up the ladder in this way until the quotient is prime. Since the quotient, [latex]3[\/latex], is prime, we stop here.\r\n\r\nDo you see why the ladder method is sometimes called stacked division?\r\nThe prime factorization is the product of all the primes on the sides and top of the ladder.\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\n<p style=\"text-align: left;\">Notice that the result is the same as we obtained with the factor tree method.<\/p>\r\n\r\n<h3 class=\"title\">Find the prime factorization of a composite number using the ladder method<\/h3>\r\n<ol id=\"eip-id1168468309839\" class=\"stepwise\">\r\n \t<li>Divide the number by the smallest prime.<\/li>\r\n \t<li>Continue dividing by that prime until it no longer divides evenly.<\/li>\r\n \t<li>Divide by the next prime until it no longer divides evenly.<\/li>\r\n \t<li>Continue until the quotient is a prime.<\/li>\r\n \t<li>Write the composite number as the product of all the primes on the sides and top of the ladder.<\/li>\r\n<\/ol>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the prime factorization of [latex]120[\/latex] using the ladder method.\r\n[reveal-answer q=\"525165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525165\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468590158\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows multiple figures of ladder division. The first figure shows the division of 2 into 120 to get the quotient 60. Two is chosen to divide into 120 because it is the smallest prime number that is a factor of 120. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division repeated, but now with another division bracket written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Since 30 is still divisible by 2 another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket. The number 15 is not divisible by 2, but it is divisible by the next prime, 3. The next figure shows the previous figure with another division bracket written around the 15 with a 3 on the outside left of the bracket and a 5 above the 15, outside of the bracket. The quotient 5 is prime, so the ladder is complete. The prime factorization of the number 120 is made up of all of the numbers from outside of the division brackets which are 2, 2, 2, 3, and 5. The prime factorization can be written as 2 times 2 times 2 times 3 times 5 or using exponents for repeated multiplication of 2 it can be written as 2 to the third power times 3 times 5.\">\r\n<tbody>\r\n<tr>\r\n<td>Divide the number by the smallest prime, which is [latex]2[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220032\/CNX_BMath_Figure_02_05_024_img-01-1.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket.\" width=\"113\" height=\"47\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue dividing by [latex]2[\/latex] until it no longer divides evenly.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220033\/CNX_BMath_Figure_02_05_024_img-03.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. Another division bracket is written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket.\" width=\"113\" height=\"96\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide by the next prime, [latex]3[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220034\/CNX_BMath_Figure_02_05_024_img-02.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. Another division bracket is written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket. Another division bracket is written around the 15 with a 3 on the outside left of the bracket and a 5 above the 15, outside of the bracket.\" width=\"113\" height=\"130\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The quotient, [latex]5[\/latex], is prime, so the ladder is complete. Write the prime factorization of [latex]120[\/latex].<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 3\\cdot 5[\/latex]\r\n\r\n[latex]{2}^{3}\\cdot 3\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCheck this yourself by multiplying the factors. The result should be [latex]120[\/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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145452&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the prime factorization of [latex]48[\/latex] using the ladder method.\r\n[reveal-answer q=\"55694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"55694\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469467686\" class=\"unnumbered unstyled\" summary=\"The image shows multiple figures of ladder division. The first figure shows the division of 2 into 48 to get the quotient 24. Two is chosen to divide into 48 because it is the smallest prime number that is a factor of 48. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division is repeated, but now with another division bracket written around the 24 with a 2 on the outside left of the bracket and a 12 above the 24, outside of the bracket. Since 12 is still divisible by 2 another division bracket is written around the 12 with a 2 on the outside left of the bracket and a 6 above the 30, outside of the bracket. The number 6 is still divisible by 2 so, another division bracket is written around the 6 with a 2 on the outside left of the bracket and a 3 above the 6, outside of the bracket. The quotient 3 is prime, so the ladder is complete. The prime factorization of the number 48 is made up of all of the numbers from outside of the division brackets which are 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>Divide the number by the smallest prime, [latex]2[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220034\/CNX_BMath_Figure_02_05_025_img-01.png\" alt=\"The image shows the division of 2 into 48 to get the quotient 24. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket.\" width=\"115\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue dividing by [latex]2[\/latex] until it no longer divides evenly.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220035\/CNX_BMath_Figure_02_05_025_img-02.png\" alt=\"The image shows the division of 2 into 48 to get the quotient 24. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket. Another division bracket is written around the 24 with a 2 on the outside left of the bracket and a 12 above the 24, outside of the bracket. Another division bracket is written around the 12 with a 2 on the outside left of the bracket and a 6 above the 12, outside of the bracket. Another division bracket is written around the 6 with a 2 on the outside left of the bracket and a 3 above the 6, outside of the bracket.\" width=\"115\" height=\"126\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The quotient, [latex]3[\/latex], is prime, so the ladder is complete. Write the prime factorization of [latex]48[\/latex].<\/td>\r\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]\r\n\r\n[latex]{2}^{4}\\cdot 3[\/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=145452&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"380\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show how to use the ladder method to find the prime factorization of two numbers.\r\n\r\nhttps:\/\/youtu.be\/V_wBWdndCuw","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the prime factorization of a number using the factor tree method<\/li>\n<li>Find the prime factorization of a number using the ladder method<\/li>\n<\/ul>\n<\/div>\n<p>In the previous section, we found the factors of a number. Prime numbers have only two factors, the number [latex]1[\/latex] and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.<\/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 same 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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end.\" width=\"293\" height=\"108\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end. Two more branches are splitting out from under 24. The left branch has the number 4 at the end and the right branch has the number 6 at the end.\" width=\"293\" height=\"181\" \/><\/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].<\/p>\n<p>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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The figure shows a factor tree with the number 48 at the top. Two branches are splitting out from under 48. The left branch has a number 2 at the end with a circle around it. The right branch has the number 24 at the end. Two more branches are splitting out from under 24. The left branch has the number 4 at the end and the right branch has the number 6 at the end. 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. Two more branches are splitting out from under 6. The left branch has the number 2 at the end with a circle around it, and the right branch has the number 3 at the end with a circle around it.\" width=\"293\" height=\"266\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The figure shows a factor tree with the number 84 at the top. Two branches are splitting out from under 84. The left branch has a number 4 at the end. The right branch has the number 21 at the end.\" width=\"311\" height=\"104\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now the factors are all prime, so we circle them.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"The figure shows a factor tree with the number 84 at the top. Two branches are splitting out from under 84. The left branch has a number 4 at the end. The right branch has the number 21 at the end. Two more branches are splitting out from under 4. The both the left and right branch has the number 2 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. Two more branches are splitting out from under 21. The left branch has the number 3 at the end with a circle around it, and the right branch has the number 7 at the end with a circle around it.\" width=\"311\" height=\"196\" \/><\/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]<\/p>\n<p>[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=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145453&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<h2>Prime Factorization Using the Ladder Method<\/h2>\n<p>The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.<\/p>\n<p>To begin building the &#8220;ladder,&#8221; divide the given number by its smallest prime factor. For example, to start the ladder for [latex]36[\/latex], we divide [latex]36[\/latex] by [latex]2[\/latex], the smallest prime factor of [latex]36[\/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\/24220030\/CNX_BMath_Figure_02_05_010_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket.\" \/><br \/>\nTo add a &#8220;step&#8221; to the ladder, we continue dividing by the same prime until it no longer divides evenly.<\/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\/24220031\/CNX_BMath_Figure_02_05_011_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket.\" \/><br \/>\nThen we divide by the next prime; so we divide [latex]9[\/latex] by [latex]3[\/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\/24220031\/CNX_BMath_Figure_02_05_012_img.png\" alt=\"The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket. Another division bracket is written around the 9 with a 3 on the outside left of the bracket and a 3 above the 9, outside of the bracket.\" \/><br \/>\nWe continue dividing up the ladder in this way until the quotient is prime. Since the quotient, [latex]3[\/latex], is prime, we stop here.<\/p>\n<p>Do you see why the ladder method is sometimes called stacked division?<br \/>\nThe prime factorization is the product of all the primes on the sides and top of the ladder.<\/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 style=\"text-align: left;\">Notice that the result is the same as we obtained with the factor tree method.<\/p>\n<h3 class=\"title\">Find the prime factorization of a composite number using the ladder method<\/h3>\n<ol id=\"eip-id1168468309839\" class=\"stepwise\">\n<li>Divide the number by the smallest prime.<\/li>\n<li>Continue dividing by that prime until it no longer divides evenly.<\/li>\n<li>Divide by the next prime until it no longer divides evenly.<\/li>\n<li>Continue until the quotient is a prime.<\/li>\n<li>Write the composite number as the product of all the primes on the sides and top of the ladder.<\/li>\n<\/ol>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the prime factorization of [latex]120[\/latex] using the ladder method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525165\">Show Solution<\/span><\/p>\n<div id=\"q525165\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468590158\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows multiple figures of ladder division. The first figure shows the division of 2 into 120 to get the quotient 60. Two is chosen to divide into 120 because it is the smallest prime number that is a factor of 120. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division repeated, but now with another division bracket written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Since 30 is still divisible by 2 another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket. The number 15 is not divisible by 2, but it is divisible by the next prime, 3. The next figure shows the previous figure with another division bracket written around the 15 with a 3 on the outside left of the bracket and a 5 above the 15, outside of the bracket. The quotient 5 is prime, so the ladder is complete. The prime factorization of the number 120 is made up of all of the numbers from outside of the division brackets which are 2, 2, 2, 3, and 5. The prime factorization can be written as 2 times 2 times 2 times 3 times 5 or using exponents for repeated multiplication of 2 it can be written as 2 to the third power times 3 times 5.\">\n<tbody>\n<tr>\n<td>Divide the number by the smallest prime, which is [latex]2[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220032\/CNX_BMath_Figure_02_05_024_img-01-1.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket.\" width=\"113\" height=\"47\" \/><\/td>\n<\/tr>\n<tr>\n<td>Continue dividing by [latex]2[\/latex] until it no longer divides evenly.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220033\/CNX_BMath_Figure_02_05_024_img-03.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. Another division bracket is written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket.\" width=\"113\" height=\"96\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide by the next prime, [latex]3[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220034\/CNX_BMath_Figure_02_05_024_img-02.png\" alt=\"The image shows the division of 2 into 120 to get the quotient 60. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. Another division bracket is written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket. Another division bracket is written around the 15 with a 3 on the outside left of the bracket and a 5 above the 15, outside of the bracket.\" width=\"113\" height=\"130\" \/><\/td>\n<\/tr>\n<tr>\n<td>The quotient, [latex]5[\/latex], is prime, so the ladder is complete. Write the prime factorization of [latex]120[\/latex].<\/td>\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 3\\cdot 5[\/latex]<\/p>\n<p>[latex]{2}^{3}\\cdot 3\\cdot 5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Check this yourself by multiplying the factors. The result should be [latex]120[\/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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145452&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the prime factorization of [latex]48[\/latex] using the ladder method.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q55694\">Show Solution<\/span><\/p>\n<div id=\"q55694\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469467686\" class=\"unnumbered unstyled\" summary=\"The image shows multiple figures of ladder division. The first figure shows the division of 2 into 48 to get the quotient 24. Two is chosen to divide into 48 because it is the smallest prime number that is a factor of 48. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division is repeated, but now with another division bracket written around the 24 with a 2 on the outside left of the bracket and a 12 above the 24, outside of the bracket. Since 12 is still divisible by 2 another division bracket is written around the 12 with a 2 on the outside left of the bracket and a 6 above the 30, outside of the bracket. The number 6 is still divisible by 2 so, another division bracket is written around the 6 with a 2 on the outside left of the bracket and a 3 above the 6, outside of the bracket. The quotient 3 is prime, so the ladder is complete. The prime factorization of the number 48 is made up of all of the numbers from outside of the division brackets which are 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>Divide the number by the smallest prime, [latex]2[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220034\/CNX_BMath_Figure_02_05_025_img-01.png\" alt=\"The image shows the division of 2 into 48 to get the quotient 24. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket.\" width=\"115\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td>Continue dividing by [latex]2[\/latex] until it no longer divides evenly.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220035\/CNX_BMath_Figure_02_05_025_img-02.png\" alt=\"The image shows the division of 2 into 48 to get the quotient 24. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket. Another division bracket is written around the 24 with a 2 on the outside left of the bracket and a 12 above the 24, outside of the bracket. Another division bracket is written around the 12 with a 2 on the outside left of the bracket and a 6 above the 12, outside of the bracket. Another division bracket is written around the 6 with a 2 on the outside left of the bracket and a 3 above the 6, outside of the bracket.\" width=\"115\" height=\"126\" \/><\/td>\n<\/tr>\n<tr>\n<td>The quotient, [latex]3[\/latex], is prime, so the ladder is complete. Write the prime factorization of [latex]48[\/latex].<\/td>\n<td>[latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>[latex]{2}^{4}\\cdot 3[\/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=145452&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"380\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show how to use the ladder method to find the prime factorization of two numbers.<\/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\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-9444\">\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>Question ID 145453, 145452, 146553, 146554. <strong>Authored by<\/strong>: Lumen Learning. <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<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/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><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><\/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 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