{"id":257660,"date":"2025-07-17T15:55:09","date_gmt":"2025-07-17T15:55:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=257660"},"modified":"2025-07-17T15:55:12","modified_gmt":"2025-07-17T15:55:12","slug":"express-132-as-the-product-of-its-prime-factors-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/17\/express-132-as-the-product-of-its-prime-factors-2\/","title":{"rendered":"Express 132 as the product of its prime factors"},"content":{"rendered":"\n

) Express 132 as the product of its prime factors. (b) The number 132k is a perfect square. Find the smallest positive integer of k. (c) p is a number between 250 and 350. The highest common factor of p and 132 is 22. Find the smallest possible value of p.<\/p>\n\n\n\n

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

Express 132 as the product of its prime factors.<\/h3>\n\n\n\n

To express 132 as the product of its prime factors, we start by dividing 132 by the smallest prime numbers.<\/p>\n\n\n\n

    \n
  1. 132 \u00f7 2 = 66<\/strong> (132 is divisible by 2, since it’s even)<\/li>\n\n\n\n
  2. 66 \u00f7 2 = 33<\/strong> (66 is divisible by 2)<\/li>\n\n\n\n
  3. 33 \u00f7 3 = 11<\/strong> (33 is divisible by 3, the sum of digits 3 + 3 = 6, which is divisible by 3)<\/li>\n\n\n\n
  4. 11 is a prime number<\/strong>, so we stop here.<\/li>\n<\/ol>\n\n\n\n

    Thus, the prime factorization of 132 is: 132=22\u00d73\u00d711132 = 2^2 \\times 3 \\times 11132=22\u00d73\u00d711<\/p>\n\n\n\n

    (b) The number 132k is a perfect square. Find the smallest positive integer of k.<\/h3>\n\n\n\n

    For 132k to be a perfect square, all the prime factors must appear with even exponents. From part (a), we know the prime factorization of 132 is: 132=22\u00d73\u00d711132 = 2^2 \\times 3 \\times 11132=22\u00d73\u00d711<\/p>\n\n\n\n

    The exponents of 3 and 11 are both 1 (odd), so we need to multiply 132 by the smallest kkk that will make all exponents even.<\/p>\n\n\n\n

    To make the exponent of 3 even, we need to multiply by another 3.
    To make the exponent of 11 even, we need to multiply by another 11.<\/p>\n\n\n\n

    Thus, the smallest value of kkk is: k=3\u00d711=33k = 3 \\times 11 = 33k=3\u00d711=33<\/p>\n\n\n\n

    Therefore, the smallest positive integer kkk is 33<\/strong>.<\/p>\n\n\n\n

    (c) p is a number between 250 and 350. The highest common factor of p and 132 is 22. Find the smallest possible value of p.<\/h3>\n\n\n\n

    We are given that the highest common factor (HCF) of ppp and 132 is 22. From part (a), the prime factorization of 132 is: 132=22\u00d73\u00d711132 = 2^2 \\times 3 \\times 11132=22\u00d73\u00d711<\/p>\n\n\n\n

    Since the HCF is 22, ppp must include the prime factors of 22, which is: 22=2\u00d71122 = 2 \\times 1122=2\u00d711<\/p>\n\n\n\n

    This means that ppp must be divisible by 2 and 11, but not by 3 (since the HCF does not include 3).<\/p>\n\n\n\n

    To find the smallest ppp between 250 and 350 that satisfies this condition, we start by checking multiples of 22 within that range.<\/p>\n\n\n\n

    The multiples of 22 between 250 and 350 are: 22\u00d712=264and22\u00d713=28622 \\times 12 = 264 \\quad \\text{and} \\quad 22 \\times 13 = 28622\u00d712=264and22\u00d713=286<\/p>\n\n\n\n

    We check their divisibility by 3:<\/p>\n\n\n\n

      \n
    • 264\u00f73=88264 \\div 3 = 88264\u00f73=88, so 264 is divisible by 3.<\/li>\n\n\n\n
    • 286\u00f73=95.33286 \\div 3 = 95.33286\u00f73=95.33, so 286 is not<\/strong> divisible by 3.<\/li>\n<\/ul>\n\n\n\n

      Thus, the smallest possible value of ppp is 286<\/strong>.<\/p>\n\n\n\n

      Final Answers:<\/h3>\n\n\n\n

      (a) The prime factorization of 132 is 22\u00d73\u00d7112^2 \\times 3 \\times 1122\u00d73\u00d711.
      (b) The smallest kkk for which 132k is a perfect square is k=33k = 33k=33.
      (c) The smallest possible value of ppp is 286.<\/p>\n\n\n\n

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

      ) Express 132 as the product of its prime factors. (b) The number 132k is a perfect square. Find the smallest positive integer of k. (c) p is a number between 250 and 350. The highest common factor of p and 132 is 22. Find the smallest possible value of p. The Correct Answer and […]<\/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-257660","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257660","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=257660"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257660\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257660"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}