{"id":197793,"date":"2025-03-07T16:27:31","date_gmt":"2025-03-07T16:27:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=197793"},"modified":"2025-03-07T16:27:34","modified_gmt":"2025-03-07T16:27:34","slug":"write-the-prime-factorization-of-each-number-as-a-product-of-powers","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/07\/write-the-prime-factorization-of-each-number-as-a-product-of-powers\/","title":{"rendered":"Write the prime factorization of each number as a product of powers"},"content":{"rendered":"\n

Write the prime factorization of each number as a product of powers. (a) 129. 3615. 1695 (b) 1613. 107.8112<\/p>\n\n\n\n

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

Prime Factorization of Each Number<\/h3>\n\n\n\n

(a) 129, 3615, 1695<\/h4>\n\n\n\n

Prime Factorization of 129:<\/strong><\/p>\n\n\n\n

    \n
  1. Start by dividing by 3 (since the sum of digits, 1 + 2 + 9 = 12, is divisible by 3):
    [
    129 \\div 3 = 43
    ]<\/li>\n\n\n\n
  2. 43 is a prime number (it is divisible only by 1 and itself).<\/li>\n<\/ol>\n\n\n\n

    Thus, the prime factorization of 129 is:
    [
    129 = 3^1 \\times 43^1
    ]<\/p>\n\n\n\n

    Prime Factorization of 3615:<\/strong><\/p>\n\n\n\n

      \n
    1. Check divisibility by 5 (since it ends in 5):
      [
      3615 \\div 5 = 723
      ]<\/li>\n\n\n\n
    2. Now, check divisibility of 723 by 3 (since the sum of digits, 7 + 2 + 3 = 12, is divisible by 3):
      [
      723 \\div 3 = 241
      ]<\/li>\n\n\n\n
    3. Check if 241 is divisible by any prime numbers. It’s not divisible by 2, 3, or 5. It turns out 241 is a prime number.<\/li>\n<\/ol>\n\n\n\n

      Thus, the prime factorization of 3615 is:
      [
      3615 = 5^1 \\times 3^1 \\times 241^1
      ]<\/p>\n\n\n\n

      Prime Factorization of 1695:<\/strong><\/p>\n\n\n\n

        \n
      1. Since the last digit is 5, we divide by 5:
        [
        1695 \\div 5 = 339
        ]<\/li>\n\n\n\n
      2. Check divisibility of 339 by 3 (since the sum of digits, 3 + 3 + 9 = 15, is divisible by 3):
        [
        339 \\div 3 = 113
        ]<\/li>\n\n\n\n
      3. 113 is a prime number.<\/li>\n<\/ol>\n\n\n\n

        Thus, the prime factorization of 1695 is:
        [
        1695 = 5^1 \\times 3^1 \\times 113^1
        ]<\/p>\n\n\n\n

        \n\n\n\n

        (b) 1613, 107, 8112<\/h4>\n\n\n\n

        Prime Factorization of 1613:<\/strong><\/p>\n\n\n\n

          \n
        1. 1613 is not divisible by 2, 3, 5, 7, or 11. Checking further primes, we find 1613 is divisible by 7:
          [
          1613 \\div 7 = 230.43
          ]
          So, 1613 is prime.<\/li>\n<\/ol>\n\n\n\n

          Thus, the prime factorization of 1613 is:
          [
          1613 = 1613^1
          ]<\/p>\n\n\n\n

          Prime Factorization of 107:<\/strong><\/p>\n\n\n\n

            \n
          1. Since 107 is not divisible by any primes up to its square root, 107 is a prime number.<\/li>\n<\/ol>\n\n\n\n

            Thus, the prime factorization of 107 is:
            [
            107 = 107^1
            ]<\/p>\n\n\n\n

            Prime Factorization of 8112:<\/strong><\/p>\n\n\n\n

              \n
            1. Since 8112 is even, divide by 2:
              [
              8112 \\div 2 = 4056
              ]
              [
              4056 \\div 2 = 2028
              ]
              [
              2028 \\div 2 = 1014
              ]
              [
              1014 \\div 2 = 507
              ]<\/li>\n\n\n\n
            2. 507 is divisible by 3 (since 5 + 0 + 7 = 12, divisible by 3):
              [
              507 \\div 3 = 169
              ]<\/li>\n\n\n\n
            3. 169 is divisible by 13:
              [
              169 \\div 13 = 13
              ]<\/li>\n\n\n\n
            4. 13 is a prime number.<\/li>\n<\/ol>\n\n\n\n

              Thus, the prime factorization of 8112 is:
              [
              8112 = 2^4 \\times 3^1 \\times 13^2
              ]<\/p>\n\n\n\n

              \n\n\n\n

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

              Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every integer greater than 1 can be expressed uniquely as a product of primes, except for the order of the factors.<\/p>\n\n\n\n

              For example:<\/p>\n\n\n\n

                \n
              • 129<\/strong> is divisible by 3 because the sum of its digits (1 + 2 + 9 = 12) is divisible by 3. After dividing, we get 43, a prime number. Hence, the prime factorization of 129 is ( 3^1 \\times 43^1 ).<\/li>\n\n\n\n
              • 3615<\/strong> is divisible by 5 because it ends in 5, then by 3 (sum of digits = 12). The result is 241, a prime number.<\/li>\n\n\n\n
              • Similarly, for 1695<\/strong>, we first divide by 5 (ends in 5) and then by 3, yielding 113, which is prime.<\/li>\n<\/ul>\n\n\n\n

                For larger numbers, like 8112<\/strong>, divisibility rules help: start by dividing by the smallest primes (2, 3, 5), and then check divisibility by other primes. This process of successive division reveals the prime factorization of a number.<\/p>\n","protected":false},"excerpt":{"rendered":"

                Write the prime factorization of each number as a product of powers. (a) 129. 3615. 1695 (b) 1613. 107.8112 The correct answer and explanation is : Prime Factorization of Each Number (a) 129, 3615, 1695 Prime Factorization of 129: Thus, the prime factorization of 129 is:[129 = 3^1 \\times 43^1] Prime Factorization of 3615: Thus, […]<\/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-197793","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197793","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=197793"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197793\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197793"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197793"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197793"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}