{"id":593,"date":"2021-08-30T12:16:23","date_gmt":"2021-08-30T12:16:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=593"},"modified":"2026-04-02T14:04:09","modified_gmt":"2026-04-02T14:04:09","slug":"1-3-operations-on-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-3-operations-on-real-numbers\/","title":{"raw":"1.3.1: Factors and the Greatest Common Factor","rendered":"1.3.1: Factors and the Greatest Common Factor"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Determine which numbers are prime and composite<\/li>\r\n \t<li>Find all the factors of a number<\/li>\r\n \t<li>Find the prime factorization of a composite number<\/li>\r\n \t<li>Find the greatest common factor of a composite number<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Factor<\/strong>: A number that divides exactly into a natural number<\/li>\r\n \t<li><strong>Prime number<\/strong>:\u00a0A number with exactly two factors, 1 and the number itself<\/li>\r\n \t<li><strong>Composite number<\/strong>:\u00a0A number with more than two factors<\/li>\r\n \t<li><strong>Prime factorization<\/strong>:\u00a0the product of prime numbers that equals a number<\/li>\r\n \t<li><strong>Greatest common factor<\/strong>:\u00a0the largest number that divides exactly into two or more numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Factors<\/h2>\r\nWhen we have the product of two numbers, we call the two numbers that are multiplied the <em><strong>factors<\/strong><\/em> of the number. We can see below that 8 and 9 are factors of 72: \u00a08 and 9 divide exactly into 72.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220012\/CNX_BMath_Figure_02_04_008_img.png\" alt=\"The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Factors<\/h3>\r\nIf [latex]a\\cdot b=m[\/latex], then [latex]a\\text{ and }b[\/latex] are factors of [latex]m[\/latex], and [latex]m[\/latex] is the product of [latex]a\\text{ and }b[\/latex].\r\n\r\n<\/div>\r\nIt can be useful to determine all of the factors of a number as it can help us solve many kinds of problems.\r\n\r\nFor example, suppose a choreographer is planning a dance for a ballet recital. There are [latex]24[\/latex] dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.\r\n\r\nIn how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of [latex]24[\/latex].\u00a0The table below\u00a0summarizes the different ways that the choreographer can arrange the dancers.\r\n<table id=\"fs-id1406287\" style=\"width: 85%;\" summary=\"The table has nine rows and three columns. The first row is a header row. The columns are labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Number of Groups<\/strong><\/th>\r\n<th><strong>Dancers per Group<\/strong><\/th>\r\n<th><strong>Total Dancers<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]1\\cdot 24=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]2\\cdot 12=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]3\\cdot 8=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]4\\cdot 6=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6\\cdot 4=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]8\\cdot 3=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]12\\cdot 2=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]24\\cdot 1=24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat patterns do you see in the table above? Did you notice that the number of groups times the number of dancers per group is always [latex]24?[\/latex] This makes sense, since there are always [latex]24[\/latex] dancers.\r\n\r\nYou may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers\u2014but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of [latex]24[\/latex], which are:\r\n<p style=\"text-align: center;\">[latex]1,2,3,4,6,8,12,24[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can find all the factors of any natural number by systematically dividing the number by each natural number, starting with [latex]1[\/latex]. If the quotient is also a natural number (meaning there is no remainder when you divide), then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Find all the factors of a counting number<\/h3>\r\n<ol id=\"eip-id1168467162605\" class=\"stepwise\">\r\n \t<li>Divide the number by each of the natural numbers, in order, until the quotient is smaller than the divisor.\r\n<ul id=\"eip-id1168263547010\">\r\n \t<li>If the quotient is a natural number, the divisor and quotient are a pair of factors.<\/li>\r\n \t<li>If the quotient is not a natural number, the divisor is not a factor.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>List all the factor pairs.<\/li>\r\n \t<li>Write all the factors in order from smallest to largest.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind all the factors of [latex]72[\/latex].\r\n\r\nSolution:\r\nDivide [latex]72[\/latex] by each of the natural numbers starting with [latex]1[\/latex]. If the quotient is a whole number, the divisor and quotient are a pair of factors.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220013\/CNX_BMath_Figure_02_04_009.png\" alt=\"The figure shows a table with the number 72 divided by numbers 1 through 8. The quotients are 72, 36, 24, 18, 14.4, 12, 10.29, and 9 respectively. \" width=\"316\" height=\"176\" \/>\r\nThe next line would have a divisor of [latex]9[\/latex] and a quotient of [latex]8[\/latex]. The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to largest, we have [latex]1,2,3,4,6,8,9,12,18,24,36,\\text{ and }72[\/latex].\r\n\r\n<\/div>\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=145439&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show how to find all the factors of [latex]30[\/latex].\r\n\r\nhttps:\/\/youtu.be\/3EL3VA2v9iI\r\n<h2>Prime and Composite Numbers<\/h2>\r\nSome numbers, like [latex]72[\/latex], have many factors. Other numbers, such as [latex]7[\/latex], have only two factors: [latex]1[\/latex] and the number. A number with only two factors is called a <em><strong>prime number<\/strong><\/em>. A number with more than two factors is called a <em><strong>composite number<\/strong><\/em>. The number [latex]1[\/latex] is neither prime nor composite. It has only one factor, itself.\r\n<div class=\"textbox shaded\">\r\n<h3>Prime Numbers and Composite Numbers<\/h3>\r\nA prime number is a counting number greater than [latex]1[\/latex] whose only factors are [latex]1[\/latex] and itself.\r\nA composite number is a counting number that is not prime.\r\n\r\n<\/div>\r\nThe table below\u00a0lists the counting numbers from [latex]2[\/latex] through [latex]20[\/latex] along with their factors. The highlighted numbers are prime, since each has only two factors.\r\n\r\nFactors of the counting numbers from [latex]2[\/latex] through [latex]20[\/latex], with prime numbers highlighted\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220015\/CNX_BMath_Figure_02_04_014.png\" alt=\"A table showing the factors of numbers from 2 to 20. \" width=\"741\" height=\"226\" \/>\r\nThe prime numbers less than [latex]20[\/latex] are [latex]2,3,5,7,11,13,17,\\text{and }19[\/latex]. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than [latex]1[\/latex] and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.\r\n<div class=\"textbox shaded\">\r\n<h3>Determine if a number is prime<\/h3>\r\n<ol id=\"eip-id1168468773932\" class=\"stepwise\">\r\n \t<li>Test each of the primes, in order, to see if it is a factor of the number.<\/li>\r\n \t<li>Start with [latex]2[\/latex] and stop when the quotient is smaller than the divisor or when a prime factor is found.<\/li>\r\n \t<li>If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify each number as prime or composite:\r\n<ol>\r\n \t<li>[latex]83[\/latex]<\/li>\r\n \t<li>[latex]77[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"242635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"242635\"]\r\n\r\nSolution:\r\n1. Test each prime, in order, to see if it is a factor of [latex]83[\/latex] , starting with [latex]2[\/latex], as shown. We will stop when the quotient is smaller than the divisor.\r\n<table id=\"fs-id3335859\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The figure shows a table with six rows and three columns. The first row is a header row and labels the rows \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Prime<\/strong><\/th>\r\n<th><strong>Test<\/strong><\/th>\r\n<th><strong>Factor of<\/strong> [latex]83?[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>Last digit of [latex]83[\/latex] is not [latex]0,2,4,6,\\text{or }8[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]8+3=11[\/latex], and [latex]11[\/latex] is not divisible by [latex]3[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>The last digit of [latex]83[\/latex] is not [latex]5[\/latex] or [latex]0[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]83\\div 7=(11.857\\dots)[\/latex]<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]11[\/latex]<\/td>\r\n<td>[latex]83\\div 11=(7.545\\dots)[\/latex]<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can stop when we get to [latex]11[\/latex] because the quotient [latex](7.545\\dots)[\/latex] is less than the divisor.\r\nWe did not find any prime numbers that are factors of [latex]83[\/latex], so we know [latex]83[\/latex] is prime.\r\n\r\n2. Test each prime, in order, to see if it is a factor of [latex]77[\/latex].\r\n<table id=\"fs-id2371111\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The figure shows a table with five rows and three columns. The first row is a header row and labels the rows \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Prime<\/strong><\/th>\r\n<th><strong>Test<\/strong><\/th>\r\n<th><strong>Factor of [latex]77?[\/latex] <\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>Last digit is not [latex]0,2,4,6,\\text{or }8[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]7+7=14[\/latex], and [latex]14[\/latex] is not divisible by [latex]3[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>the last digit is not [latex]5[\/latex] or [latex]0[\/latex].<\/td>\r\n<td>No.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]77\\div 11=7[\/latex]<\/td>\r\n<td>Yes.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]77[\/latex] is divisible by [latex]7[\/latex], we know it is not a prime number. It is composite.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145441[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to determine whether a number is prime or composite.\r\n\r\nhttps:\/\/youtu.be\/8v7baCT33xw\r\n<h2>Prime Factorization<\/h2>\r\nPrime 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 <em><strong>prime factorization<\/strong><\/em> 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<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), as we cannot 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<div class=\"textbox shaded\">\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>\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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it.\" 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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it. Two more branches are splitting out from under 24. The right branch has the number 6 at the end and the left branch has the number 4 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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it. Two more branches are splitting out from under 24. The right branch has the number 6 at the end and the left branch has the number 4 at the end . Two more branches are splitting out from under 6. The left and right branch have the number 2 and 3 at the end with a circle around it. Another 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.\" 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<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<th>Process<\/th>\r\n<th>Factor Tree<\/th>\r\n<\/tr>\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 right branch has a number 21 at the end. The left branch has the number 4 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 48. The right branch has a number 21 at the end. The left branch has the number 4 at the end. Two more branches are splitting out from under 21. The right branch has the number 7 at the end with a circle around it 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 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<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\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Finding the Greatest Common Factor<\/h2>\r\nThe <em><strong>greatest common factor<\/strong><\/em> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex]\u00a0is the GCF of [latex]16[\/latex]\u00a0and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex]\u00a0and [latex]20[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Greatest Common Factor<\/h3>\r\nThe greatest common factor (GCF) of two or more natural numbers is the largest natural number that is a factor of all the numbers.\r\n\r\n<\/div>\r\nWe will show how to find the greatest common factor of two numbers.\r\n\r\nLet\u2019s first find the <em><strong>greatest common factor (GCF)<\/strong><\/em> of two whole numbers. The GCF of two natural numbers is the greatest natural number that is a factor of <i>both<\/i> of the numbers. Take the numbers [latex]50[\/latex] and [latex]30[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}50=10\\cdot5\\\\30=10\\cdot3\\end{array}[\/latex]<\/p>\r\nTheir greatest common factor is [latex]10[\/latex], since [latex]10[\/latex] is the greatest factor that both numbers have in common.\r\n\r\nTo find the GCF of larger numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says, \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 194px;\" colspan=\"2\"><strong>Process<\/strong><\/th>\r\n<th style=\"width: 426.45px;\">Factor Tree<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\r\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\r\n<td style=\"width: 426.45px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\"Two factor trees for the number 24 and 36. The end factors of 24 are 2, 2, 2, and 3. The end factors of 36 are 2, 2, 3, and 3.\" width=\"460\" height=\"193\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors--matching common factors in a column.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\"Two mathematical sentences. The first sentence says that 24 is equal to 2 times 2 times 2 times 3. The second sentence says that 36 is equal to 2 times 2 times 3 times 3. \" width=\"459\" height=\"65\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\r\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\r\n<td style=\"width: 426.45px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\"Two mathematical sentences. The first sentence says that 24 is equal to 2 times 2 times 2 times 3. The second sentence says that 36 is equal to 2 times 2 times 3 times 3. The common factors 2, 2 and 3 in each sentence are circled. The GCF is equal to 2 times 2 times 3, which is equal to 12. \" width=\"459\" height=\"141\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\r\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\r\n<td style=\"width: 426.45px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\r\n<td style=\"width: 199.55px;\"><\/td>\r\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].\r\n\r\n[reveal-answer q=\"803757\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803757\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\text{GCF}=42[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and[latex]168[\/latex] by [latex]42[\/latex]\u2014they are both exactly divisible by this number!)<\/div>\r\nThe video that follows shows another example of finding the greatest common factor of two natural numbers.\r\n\r\nhttps:\/\/youtu.be\/KbBJcdDY_VE\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146326[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>The Greatest Common Factor of More Than Two Numbers<\/h2>\r\nThe greatest common factor of more than two numbers is found in a similar way. The prime numbers used for the GCF must be factor triple, quadruples etc.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the greatest common factor of [latex]48, 36, \\text{and }30[\/latex].\r\n<h4><strong>Solution<\/strong><\/h4>\r\nFirst write the prime factorization for each number:\r\n\r\n[latex]48=2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]\r\n\r\n[latex]36=2\\cdot 2\\cdot 3\\cdot 3[\/latex]\r\n\r\n[latex]30=2\\cdot 3\\cdot 5[\/latex]\r\n\r\nNow line up the prime factorizations into columns of the same prime, determine which primes are common to all 3 numbers, multiply these primes to find the GFC.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the greatest common factor of [latex]77, 154, 63, \\text{and }126[\/latex].\r\n\r\n[reveal-answer q=\"619876\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"619876\"]\r\n\r\n[latex]77=7\\cdot 11[\/latex]\r\n\r\n[latex]154=2\\cdot 7\\cdot 11[\/latex]\r\n\r\n[latex]63=3\\cdot 3\\cdot 7[\/latex]\r\n\r\n[latex]126=2\\cdot 3\\cdot 3\\cdot 7\\cdot[\/latex]\r\n\r\n[latex]\\text{GCF }=7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Determine which numbers are prime and composite<\/li>\n<li>Find all the factors of a number<\/li>\n<li>Find the prime factorization of a composite number<\/li>\n<li>Find the greatest common factor of a composite number<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Factor<\/strong>: A number that divides exactly into a natural number<\/li>\n<li><strong>Prime number<\/strong>:\u00a0A number with exactly two factors, 1 and the number itself<\/li>\n<li><strong>Composite number<\/strong>:\u00a0A number with more than two factors<\/li>\n<li><strong>Prime factorization<\/strong>:\u00a0the product of prime numbers that equals a number<\/li>\n<li><strong>Greatest common factor<\/strong>:\u00a0the largest number that divides exactly into two or more numbers<\/li>\n<\/ul>\n<\/div>\n<h2>Factors<\/h2>\n<p>When we have the product of two numbers, we call the two numbers that are multiplied the <em><strong>factors<\/strong><\/em> of the number. We can see below that 8 and 9 are factors of 72: \u00a08 and 9 divide exactly into 72.<\/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\/24220012\/CNX_BMath_Figure_02_04_008_img.png\" alt=\"The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Factors<\/h3>\n<p>If [latex]a\\cdot b=m[\/latex], then [latex]a\\text{ and }b[\/latex] are factors of [latex]m[\/latex], and [latex]m[\/latex] is the product of [latex]a\\text{ and }b[\/latex].<\/p>\n<\/div>\n<p>It can be useful to determine all of the factors of a number as it can help us solve many kinds of problems.<\/p>\n<p>For example, suppose a choreographer is planning a dance for a ballet recital. There are [latex]24[\/latex] dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.<\/p>\n<p>In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of [latex]24[\/latex].\u00a0The table below\u00a0summarizes the different ways that the choreographer can arrange the dancers.<\/p>\n<table id=\"fs-id1406287\" style=\"width: 85%;\" summary=\"The table has nine rows and three columns. The first row is a header row. The columns are labeled\">\n<thead>\n<tr valign=\"top\">\n<th><strong>Number of Groups<\/strong><\/th>\n<th><strong>Dancers per Group<\/strong><\/th>\n<th><strong>Total Dancers<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]1\\cdot 24=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]2\\cdot 12=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]3\\cdot 8=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]4\\cdot 6=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6\\cdot 4=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8\\cdot 3=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]12\\cdot 2=24[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]24\\cdot 1=24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What patterns do you see in the table above? Did you notice that the number of groups times the number of dancers per group is always [latex]24?[\/latex] This makes sense, since there are always [latex]24[\/latex] dancers.<\/p>\n<p>You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers\u2014but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of [latex]24[\/latex], which are:<\/p>\n<p style=\"text-align: center;\">[latex]1,2,3,4,6,8,12,24[\/latex]<\/p>\n<p style=\"text-align: left;\">We can find all the factors of any natural number by systematically dividing the number by each natural number, starting with [latex]1[\/latex]. If the quotient is also a natural number (meaning there is no remainder when you divide), then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.<\/p>\n<div class=\"textbox shaded\">\n<h3>Find all the factors of a counting number<\/h3>\n<ol id=\"eip-id1168467162605\" class=\"stepwise\">\n<li>Divide the number by each of the natural numbers, in order, until the quotient is smaller than the divisor.\n<ul id=\"eip-id1168263547010\">\n<li>If the quotient is a natural number, the divisor and quotient are a pair of factors.<\/li>\n<li>If the quotient is not a natural number, the divisor is not a factor.<\/li>\n<\/ul>\n<\/li>\n<li>List all the factor pairs.<\/li>\n<li>Write all the factors in order from smallest to largest.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find all the factors of [latex]72[\/latex].<\/p>\n<p>Solution:<br \/>\nDivide [latex]72[\/latex] by each of the natural numbers starting with [latex]1[\/latex]. If the quotient is a whole number, the divisor and quotient are a pair of factors.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220013\/CNX_BMath_Figure_02_04_009.png\" alt=\"The figure shows a table with the number 72 divided by numbers 1 through 8. The quotients are 72, 36, 24, 18, 14.4, 12, 10.29, and 9 respectively.\" width=\"316\" height=\"176\" \/><br \/>\nThe next line would have a divisor of [latex]9[\/latex] and a quotient of [latex]8[\/latex]. The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to largest, we have [latex]1,2,3,4,6,8,9,12,18,24,36,\\text{ and }72[\/latex].<\/p>\n<\/div>\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=145439&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show how to find all the factors of [latex]30[\/latex].<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine Factors of a Number\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3EL3VA2v9iI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Prime and Composite Numbers<\/h2>\n<p>Some numbers, like [latex]72[\/latex], have many factors. Other numbers, such as [latex]7[\/latex], have only two factors: [latex]1[\/latex] and the number. A number with only two factors is called a <em><strong>prime number<\/strong><\/em>. A number with more than two factors is called a <em><strong>composite number<\/strong><\/em>. The number [latex]1[\/latex] is neither prime nor composite. It has only one factor, itself.<\/p>\n<div class=\"textbox shaded\">\n<h3>Prime Numbers and Composite Numbers<\/h3>\n<p>A prime number is a counting number greater than [latex]1[\/latex] whose only factors are [latex]1[\/latex] and itself.<br \/>\nA composite number is a counting number that is not prime.<\/p>\n<\/div>\n<p>The table below\u00a0lists the counting numbers from [latex]2[\/latex] through [latex]20[\/latex] along with their factors. The highlighted numbers are prime, since each has only two factors.<\/p>\n<p>Factors of the counting numbers from [latex]2[\/latex] through [latex]20[\/latex], with prime numbers highlighted<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220015\/CNX_BMath_Figure_02_04_014.png\" alt=\"A table showing the factors of numbers from 2 to 20.\" width=\"741\" height=\"226\" \/><br \/>\nThe prime numbers less than [latex]20[\/latex] are [latex]2,3,5,7,11,13,17,\\text{and }19[\/latex]. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than [latex]1[\/latex] and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.<\/p>\n<div class=\"textbox shaded\">\n<h3>Determine if a number is prime<\/h3>\n<ol id=\"eip-id1168468773932\" class=\"stepwise\">\n<li>Test each of the primes, in order, to see if it is a factor of the number.<\/li>\n<li>Start with [latex]2[\/latex] and stop when the quotient is smaller than the divisor or when a prime factor is found.<\/li>\n<li>If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify each number as prime or composite:<\/p>\n<ol>\n<li>[latex]83[\/latex]<\/li>\n<li>[latex]77[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q242635\">Show Solution<\/span><\/p>\n<div id=\"q242635\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. Test each prime, in order, to see if it is a factor of [latex]83[\/latex] , starting with [latex]2[\/latex], as shown. We will stop when the quotient is smaller than the divisor.<\/p>\n<table id=\"fs-id3335859\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The figure shows a table with six rows and three columns. The first row is a header row and labels the rows\">\n<thead>\n<tr valign=\"top\">\n<th><strong>Prime<\/strong><\/th>\n<th><strong>Test<\/strong><\/th>\n<th><strong>Factor of<\/strong> [latex]83?[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>Last digit of [latex]83[\/latex] is not [latex]0,2,4,6,\\text{or }8[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8+3=11[\/latex], and [latex]11[\/latex] is not divisible by [latex]3[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]5[\/latex]<\/td>\n<td>The last digit of [latex]83[\/latex] is not [latex]5[\/latex] or [latex]0[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]83\\div 7=(11.857\\dots)[\/latex]<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]11[\/latex]<\/td>\n<td>[latex]83\\div 11=(7.545\\dots)[\/latex]<\/td>\n<td>No.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can stop when we get to [latex]11[\/latex] because the quotient [latex](7.545\\dots)[\/latex] is less than the divisor.<br \/>\nWe did not find any prime numbers that are factors of [latex]83[\/latex], so we know [latex]83[\/latex] is prime.<\/p>\n<p>2. Test each prime, in order, to see if it is a factor of [latex]77[\/latex].<\/p>\n<table id=\"fs-id2371111\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"The figure shows a table with five rows and three columns. The first row is a header row and labels the rows\">\n<thead>\n<tr valign=\"top\">\n<th><strong>Prime<\/strong><\/th>\n<th><strong>Test<\/strong><\/th>\n<th><strong>Factor of [latex]77?[\/latex] <\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>Last digit is not [latex]0,2,4,6,\\text{or }8[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]7+7=14[\/latex], and [latex]14[\/latex] is not divisible by [latex]3[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]5[\/latex]<\/td>\n<td>the last digit is not [latex]5[\/latex] or [latex]0[\/latex].<\/td>\n<td>No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]77\\div 11=7[\/latex]<\/td>\n<td>Yes.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]77[\/latex] is divisible by [latex]7[\/latex], we know it is not a prime number. It is composite.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145441&theme=oea&iframe_resize_id=ohm145441&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to determine whether a number is prime or composite.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determine if Numbers Are Prime or Composite (Algorithm)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8v7baCT33xw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Prime Factorization<\/h2>\n<p>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 <em><strong>prime factorization<\/strong><\/em> 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<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), as we cannot 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<div class=\"textbox shaded\">\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>\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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it.\" 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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it. Two more branches are splitting out from under 24. The right branch has the number 6 at the end and the left branch has the number 4 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 right branch has a number 24 at the end. The left branch has the number 2 at the end with a circle around it. Two more branches are splitting out from under 24. The right branch has the number 6 at the end and the left branch has the number 4 at the end . Two more branches are splitting out from under 6. The left and right branch have the number 2 and 3 at the end with a circle around it. Another 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.\" 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<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-3\" 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<th>Process<\/th>\n<th>Factor Tree<\/th>\n<\/tr>\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 right branch has a number 21 at the end. The left branch has the number 4 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 48. The right branch has a number 21 at the end. The left branch has the number 4 at the end. Two more branches are splitting out from under 21. The right branch has the number 7 at the end with a circle around it 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 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<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\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Greatest Common Factor<\/h2>\n<p>The <em><strong>greatest common factor<\/strong><\/em> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex]\u00a0is the GCF of [latex]16[\/latex]\u00a0and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex]\u00a0and [latex]20[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Greatest Common Factor<\/h3>\n<p>The greatest common factor (GCF) of two or more natural numbers is the largest natural number that is a factor of all the numbers.<\/p>\n<\/div>\n<p>We will show how to find the greatest common factor of two numbers.<\/p>\n<p>Let\u2019s first find the <em><strong>greatest common factor (GCF)<\/strong><\/em> of two whole numbers. The GCF of two natural numbers is the greatest natural number that is a factor of <i>both<\/i> of the numbers. Take the numbers [latex]50[\/latex] and [latex]30[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}50=10\\cdot5\\\\30=10\\cdot3\\end{array}[\/latex]<\/p>\n<p>Their greatest common factor is [latex]10[\/latex], since [latex]10[\/latex] is the greatest factor that both numbers have in common.<\/p>\n<p>To find the GCF of larger numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the greatest common factor of [latex]24[\/latex] and [latex]36[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168464918810\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Three columns are shown. The top row of the first column says,\">\n<tbody>\n<tr>\n<th style=\"width: 194px;\" colspan=\"2\"><strong>Process<\/strong><\/th>\n<th style=\"width: 426.45px;\">Factor Tree<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 1:<\/strong> Factor each coefficient into primes. Write all variables with exponents in expanded form.<\/td>\n<td style=\"width: 199.55px;\">Factor [latex]24[\/latex] and [latex]36[\/latex].<\/td>\n<td style=\"width: 426.45px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224611\/CNX_BMath_Figure_10_06_024_img-01.png\" alt=\"Two factor trees for the number 24 and 36. The end factors of 24 are 2, 2, 2, and 3. The end factors of 36 are 2, 2, 3, and 3.\" width=\"460\" height=\"193\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 2:<\/strong> List all factors&#8211;matching common factors in a column.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224614\/CNX_BMath_Figure_10_06_024_img-02.png\" alt=\"Two mathematical sentences. The first sentence says that 24 is equal to 2 times 2 times 2 times 3. The second sentence says that 36 is equal to 2 times 2 times 3 times 3.\" width=\"459\" height=\"65\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\">In each column, circle the common factors.<\/td>\n<td style=\"width: 199.55px;\">Circle the [latex]2, 2[\/latex], and [latex]3[\/latex] that are shared by both numbers.<\/td>\n<td style=\"width: 426.45px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224615\/CNX_BMath_Figure_10_06_024_img-03.png\" alt=\"Two mathematical sentences. The first sentence says that 24 is equal to 2 times 2 times 2 times 3. The second sentence says that 36 is equal to 2 times 2 times 3 times 3. The common factors 2, 2 and 3 in each sentence are circled. The GCF is equal to 2 times 2 times 3, which is equal to 12.\" width=\"459\" height=\"141\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 3:<\/strong> Bring down the common factors that all expressions share.<\/td>\n<td style=\"width: 199.55px;\">Bring down the [latex]2, 2, 3[\/latex] and then multiply.<\/td>\n<td style=\"width: 426.45px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 194px;\"><strong>Step 4:<\/strong> Multiply the factors.<\/td>\n<td style=\"width: 199.55px;\"><\/td>\n<td style=\"width: 426.45px;\">The GCF of [latex]24[\/latex] and [latex]36[\/latex] is [latex]12[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that since the GCF is a factor of both numbers, [latex]24[\/latex] and [latex]36[\/latex] can be written as multiples of [latex]12[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}24=12\\cdot 2\\\\ 36=12\\cdot 3\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]210[\/latex] and [latex]168[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803757\">Show Solution<\/span><\/p>\n<div id=\"q803757\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,210=2\\cdot3\\cdot5\\cdot7\\\\\\,\\,\\,\\,168=2\\cdot2\\cdot2\\cdot3\\cdot7\\\\\\text{GCF}=2\\cdot3\\cdot7\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\text{GCF}=42[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both [latex]210[\/latex] and[latex]168[\/latex] by [latex]42[\/latex]\u2014they are both exactly divisible by this number!)<\/div>\n<p>The video that follows shows another example of finding the greatest common factor of two natural numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Example:  Determining the Greatest Common Factor\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KbBJcdDY_VE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146326&theme=oea&iframe_resize_id=ohm146326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>The Greatest Common Factor of More Than Two Numbers<\/h2>\n<p>The greatest common factor of more than two numbers is found in a similar way. The prime numbers used for the GCF must be factor triple, quadruples etc.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the greatest common factor of [latex]48, 36, \\text{and }30[\/latex].<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>First write the prime factorization for each number:<\/p>\n<p>[latex]48=2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>[latex]36=2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>[latex]30=2\\cdot 3\\cdot 5[\/latex]<\/p>\n<p>Now line up the prime factorizations into columns of the same prime, determine which primes are common to all 3 numbers, multiply these primes to find the GFC.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the greatest common factor of [latex]77, 154, 63, \\text{and }126[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q619876\">Show Answer<\/span><\/p>\n<div id=\"q619876\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]77=7\\cdot 11[\/latex]<\/p>\n<p>[latex]154=2\\cdot 7\\cdot 11[\/latex]<\/p>\n<p>[latex]63=3\\cdot 3\\cdot 7[\/latex]<\/p>\n<p>[latex]126=2\\cdot 3\\cdot 3\\cdot 7\\cdot[\/latex]<\/p>\n<p>[latex]\\text{GCF }=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-593\">\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 145433, 145363, 145413, 145417, 145418.. <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><li>The Greatest Common Factor of More Than Two Numbers. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Determine the Multiples of a Given Number. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mkEWqspRVKk\">https:\/\/youtu.be\/mkEWqspRVKk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Divisibility Rules. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/i16N01IdIhk\">https:\/\/youtu.be\/i16N01IdIhk<\/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>Determine if Numbers Are Prime or Composite (Algorithm). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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>Authored 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":422605,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 145433, 145363, 145413, 145417, 145418.\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License\"},{\"type\":\"cc\",\"description\":\"Determine the Multiples of a Given Number\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\" https:\/\/youtu.be\/mkEWqspRVKk\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Divisibility Rules\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/i16N01IdIhk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"OpenStax\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\" Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Determine if Numbers Are Prime or Composite (Algorithm)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"The Greatest Common Factor of More Than Two Numbers\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-593","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/593","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/593\/revisions"}],"predecessor-version":[{"id":3227,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/593\/revisions\/3227"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/593\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=593"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=593"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=593"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=593"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}