{"id":259862,"date":"2025-07-19T05:56:25","date_gmt":"2025-07-19T05:56:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259862"},"modified":"2025-07-19T05:56:28","modified_gmt":"2025-07-19T05:56:28","slug":"express-60-as-a-product-of-its-prime-factors","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/express-60-as-a-product-of-its-prime-factors\/","title":{"rendered":"Express 60 as a product of its prime factors"},"content":{"rendered":"\n

Express 60 as a product of its prime factors. Find the smallest possible integer k such that 60k is a square number.<\/p>\n\n\n\n

The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n

To express 60 as a product of its prime factors, we start by dividing 60 by the smallest prime numbers.<\/p>\n\n\n\n

    \n
  1. 60 is divisible by 2 (since it is even):
    60\u00f72=3060 \u00f7 2 = 30<\/li>\n\n\n\n
  2. 30 is divisible by 2 again:
    30\u00f72=1530 \u00f7 2 = 15<\/li>\n\n\n\n
  3. 15 is divisible by 3 (since the sum of the digits, 1 + 5 = 6, is divisible by 3):
    15\u00f73=515 \u00f7 3 = 5<\/li>\n\n\n\n
  4. 5 is a prime number.<\/li>\n<\/ol>\n\n\n\n

    Thus, the prime factorization of 60 is:
    60=22\u00d73\u00d7560 = 2^2 \u00d7 3 \u00d7 5<\/p>\n\n\n\n

    To find the smallest integer kk such that 60k60k is a square number:<\/h3>\n\n\n\n

    A square number is a number that can be expressed as the product of identical factors, meaning each prime factor’s exponent in its prime factorization must be even.<\/p>\n\n\n\n

    The prime factorization of 60 is 22\u00d731\u00d7512^2 \u00d7 3^1 \u00d7 5^1. To make 60k60k a perfect square, the exponents of all the primes must be even:<\/p>\n\n\n\n

      \n
    • The exponent of 2 is already even (222^2).<\/li>\n\n\n\n
    • The exponent of 3 is odd (313^1), so we need to multiply by one more 3 to make it even.<\/li>\n\n\n\n
    • The exponent of 5 is odd (515^1), so we need to multiply by one more 5 to make it even.<\/li>\n<\/ul>\n\n\n\n

      Thus, we multiply 60 by 3\u00d75=153 \u00d7 5 = 15 to make 60k60k a square number. Therefore, the smallest integer kk is:<\/p>\n\n\n\n

      k=15k = 15<\/p>\n\n\n\n

      Now, multiplying 60\u00d71560 \\times 15 gives:<\/p>\n\n\n\n

      60\u00d715=90060 \\times 15 = 900<\/p>\n\n\n\n

      The prime factorization of 900 is:<\/p>\n\n\n\n

      900=22\u00d732\u00d752900 = 2^2 \u00d7 3^2 \u00d7 5^2<\/p>\n\n\n\n

      Since all the exponents are even, 900 is a perfect square. Thus, the smallest kk is 15, and 60k60k is now a square number.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Express 60 as a product of its prime factors. Find the smallest possible integer k such that 60k is a square number. The correct answer and explanation is: To express 60 as a product of its prime factors, we start by dividing 60 by the smallest prime numbers. Thus, the prime factorization of 60 is:60=22\u00d73\u00d7560 […]<\/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-259862","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259862","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=259862"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259862\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259862"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}