{"id":230811,"date":"2025-06-10T06:28:45","date_gmt":"2025-06-10T06:28:45","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=230811"},"modified":"2025-06-10T06:28:47","modified_gmt":"2025-06-10T06:28:47","slug":"find-the-lcm-by-prime-factorization-of-45-75-and-125","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/10\/find-the-lcm-by-prime-factorization-of-45-75-and-125\/","title":{"rendered":"Find the LCM by prime factorization of 45, 75, and 125."},"content":{"rendered":"\n

Find the LCM by prime factorization of 45, 75, and 125.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Answer:<\/strong><\/h3>\n\n\n\n

To find the Least Common Multiple (LCM)<\/strong> of 45<\/strong>, 75<\/strong>, and 125<\/strong> using prime factorization<\/strong>, follow these steps:<\/p>\n\n\n\n

\n\n\n\n

Step 1: Prime factorize each number.<\/strong><\/h4>\n\n\n\n
    \n
  • 45=32\u00d7545 = 3^2 \\times 545=32\u00d75<\/li>\n\n\n\n
  • 75=3\u00d75275 = 3 \\times 5^275=3\u00d752<\/li>\n\n\n\n
  • 125=53125 = 5^3125=53<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 2: Take the highest power of each prime factor.<\/strong><\/h4>\n\n\n\n
      \n
    • For 3<\/strong>, the highest power is 323^232<\/li>\n\n\n\n
    • For 5<\/strong>, the highest power is 535^353<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Step 3: Multiply the highest powers together.<\/strong><\/h4>\n\n\n\n

      LCM=32\u00d753=9\u00d7125=1125\\text{LCM} = 3^2 \\times 5^3 = 9 \\times 125 = 1125LCM=32\u00d753=9\u00d7125=1125<\/p>\n\n\n\n

      \n\n\n\n

      \u2705 LCM(45, 75, 125) = 1125<\/strong><\/h3>\n\n\n\n
      \n\n\n\n

      Explanation<\/strong><\/h3>\n\n\n\n

      The Least Common Multiple (LCM)<\/strong> of a set of numbers is the smallest positive number that is divisible by all the numbers<\/strong> in the set. One effective method to find the LCM is by using prime factorization<\/strong>, which involves breaking each number down into its basic prime factors.<\/p>\n\n\n\n

      Let\u2019s apply this to 45, 75, and 125:<\/p>\n\n\n\n

        \n
      • 45<\/strong> is divisible by 3 and 5. Its prime factorization is 3\u00d73\u00d75=32\u00d753 \\times 3 \\times 5 = 3^2 \\times 53\u00d73\u00d75=32\u00d75.<\/li>\n\n\n\n
      • 75<\/strong> also breaks into primes 3 and 5. 75=3\u00d75\u00d75=3\u00d75275 = 3 \\times 5 \\times 5 = 3 \\times 5^275=3\u00d75\u00d75=3\u00d752.<\/li>\n\n\n\n
      • 125<\/strong> is a power of 5. 125=5\u00d75\u00d75=53125 = 5 \\times 5 \\times 5 = 5^3125=5\u00d75\u00d75=53.<\/li>\n<\/ul>\n\n\n\n

        To find the LCM, we look at each distinct prime factor<\/strong> that appears in any of the numbers and use the highest exponent<\/strong> that occurs for each.<\/p>\n\n\n\n

          \n
        • The highest power of 3<\/strong> among the factorizations is 323^232 (from 45).<\/li>\n\n\n\n
        • The highest power of 5<\/strong> is 535^353 (from 125).<\/li>\n<\/ul>\n\n\n\n

          Multiplying these together gives the LCM:32\u00d753=9\u00d7125=11253^2 \\times 5^3 = 9 \\times 125 = 112532\u00d753=9\u00d7125=1125<\/p>\n\n\n\n

          So, 1125<\/strong> is the smallest number that is a multiple of 45, 75, and 125.<\/p>\n\n\n\n

          This method ensures we don\u2019t miss any necessary factors, which can happen with other methods like listing multiples. Prime factorization is especially useful for larger or more complex numbers and is a foundational skill in number theory and problem-solving.<\/p>\n\n\n\n

          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          Find the LCM by prime factorization of 45, 75, and 125. The Correct Answer and Explanation is: Answer: To find the Least Common Multiple (LCM) of 45, 75, and 125 using prime factorization, follow these steps: Step 1: Prime factorize each number. Step 2: Take the highest power of each prime factor. Step 3: Multiply […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-230811","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230811","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=230811"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230811\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=230811"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=230811"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=230811"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}