{"id":6349,"date":"2017-05-02T16:05:28","date_gmt":"2017-05-02T16:05:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=6349"},"modified":"2020-01-09T00:14:19","modified_gmt":"2020-01-09T00:14:19","slug":"prime-factorization-and-the-least-common-multiple","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/prime-factorization-and-the-least-common-multiple\/","title":{"raw":"Introduction to Prime Factorization and the Least Common Multiple","rendered":"Introduction to Prime Factorization and the Least Common Multiple"},"content":{"raw":"<h2>What you'll learn to do: Use prime factorization to find the least common multiple of a number<\/h2>\r\n[caption id=\"attachment_13125\" align=\"aligncenter\" width=\"802\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/23215650\/2470743749_a92e32b075_o.jpg\"><img class=\"wp-image-13125 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/23215650\/2470743749_a92e32b075_o-1024x768.jpg\" alt=\"A modern-looking passenger train at a station platform\" width=\"802\" height=\"602\" \/><\/a> When will the train arrive?[\/caption]\r\n\r\nPeter is exploring a new city, and he's getting around by train. There are three\u00a0train lines that leave from the station closest to his hostel. One arrives\u00a0every [latex]15[\/latex] minutes, one arrives\u00a0every [latex]12[\/latex] minutes, and one arrives\u00a0every [latex]9[\/latex] minutes. If all the trains depart the station at the same time every morning, how long will it be before they're all at the station at the same time again? To find this out, you'll use prime factorization and find the least common multiple--we'll explore both of those concepts in this section.\r\n\r\nBefore you get started in this module, try a few practice problems and review\u00a0prior\u00a0concepts.\r\n<div class=\"textbox examples\">\r\n<h3>readiness quiz<\/h3>\r\n1.\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145433&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n2. Is [latex]810[\/latex] divisible by [latex]2,3,5,6,\\text{ or }10?[\/latex]\r\n\r\nAnswer: [latex]2, 3, 5, 6,\\text{ and }10[\/latex]\r\n\r\nIf you missed this problem, review the following video.\r\n\r\nhttps:\/\/youtu.be\/8A8sGvn0AeA\r\n\r\n&nbsp;\r\n\r\n3. [ohm_question]148234[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n4.\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145441&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe>\r\nIf you missed this problem, review the following example.\r\n<div class=\"textbox shaded\">\r\n\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=\"780523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"780523\"]\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\\ldots[\/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\\ldots[\/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]\\text{(7.545}\\ldots\\text{)}[\/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\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>","rendered":"<h2>What you&#8217;ll learn to do: Use prime factorization to find the least common multiple of a number<\/h2>\n<div id=\"attachment_13125\" style=\"width: 812px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/23215650\/2470743749_a92e32b075_o.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-13125\" class=\"wp-image-13125\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/23215650\/2470743749_a92e32b075_o-1024x768.jpg\" alt=\"A modern-looking passenger train at a station platform\" width=\"802\" height=\"602\" \/><\/a><\/p>\n<p id=\"caption-attachment-13125\" class=\"wp-caption-text\">When will the train arrive?<\/p>\n<\/div>\n<p>Peter is exploring a new city, and he&#8217;s getting around by train. There are three\u00a0train lines that leave from the station closest to his hostel. One arrives\u00a0every [latex]15[\/latex] minutes, one arrives\u00a0every [latex]12[\/latex] minutes, and one arrives\u00a0every [latex]9[\/latex] minutes. If all the trains depart the station at the same time every morning, how long will it be before they&#8217;re all at the station at the same time again? To find this out, you&#8217;ll use prime factorization and find the least common multiple&#8211;we&#8217;ll explore both of those concepts in this section.<\/p>\n<p>Before you get started in this module, try a few practice problems and review\u00a0prior\u00a0concepts.<\/p>\n<div class=\"textbox examples\">\n<h3>readiness quiz<\/h3>\n<p>1.<br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145433&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p>2. Is [latex]810[\/latex] divisible by [latex]2,3,5,6,\\text{ or }10?[\/latex]<\/p>\n<p>Answer: [latex]2, 3, 5, 6,\\text{ and }10[\/latex]<\/p>\n<p>If you missed this problem, review the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Apply Divisibility Rules to a 4 Digit Number\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8A8sGvn0AeA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>3. <iframe loading=\"lazy\" id=\"ohm148234\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=148234&theme=oea&iframe_resize_id=ohm148234&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>4.<br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145441&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe><br \/>\nIf you missed this problem, review the following example.<\/p>\n<div class=\"textbox shaded\">\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=\"q780523\">Show Solution<\/span><\/p>\n<div id=\"q780523\" 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\\ldots[\/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\\ldots[\/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]\\text{(7.545}\\ldots\\text{)}[\/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.<br \/>\n2. 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>\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-6349\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Apply Divisibility Rules to a 4 Digit Number. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8A8sGvn0AeA\">https:\/\/youtu.be\/8A8sGvn0AeA<\/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>Train at a station. <strong>Authored by<\/strong>: harlock81. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.flickr.com\/photos\/harlock81\/2470743749\/\">https:\/\/www.flickr.com\/photos\/harlock81\/2470743749\/<\/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>Question ID: 145433, 145411. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Apply Divisibility Rules to a 4 Digit Number\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/8A8sGvn0AeA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Train at a 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