{"id":8,"date":"2017-08-23T06:41:22","date_gmt":"2017-08-23T06:41:22","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=8"},"modified":"2017-09-18T13:34:26","modified_gmt":"2017-09-18T13:34:26","slug":"fundamental-theorem-of-arithmetic","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/fundamental-theorem-of-arithmetic\/","title":{"rendered":"Fundamental Theorem of Arithmetic"},"content":{"rendered":"
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\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Fundamental Theorem of Arithmetic<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study and apply the Fundamental Theorem of Arithmetic and theoretic number concepts of\u00a0primes, factors, and multiples.<\/p>\n

Previously Covered:<\/h4>\n
    \n
  1. A prime number<\/strong><\/em> is a number divisible only by 1 and itself.<\/li>\n
  2. A composite number<\/strong><\/em> is a number with factors other than 1 and itself.<\/li>\n<\/ol>\n
    \n

    What is the Fundamental Theorem of Arithmetic?<\/h3>\n

    The Fundamental Theorem of Arithmetic<\/strong><\/em> states that every positive integer (except the\u00a0number 1) can\u00a0be represented in exactly one way apart from rearrangement as a product of one or more prime numbers. Basically,\u00a0this means that any number (except 1) can be broken down into its prime factorization<\/strong><\/em>.<\/p>\n

    How do I find the prime factorization of a number?<\/strong><\/p>\n

    The prime factorization of a number is made up of all the numbers’ divisors that are prime numbers.<\/p>\n

    To find the prime factorization of a number, create a factor tree.<\/p>\n

    \"\"<\/p>\n

    The prime factorization of 250 is 2 \u00b7 5 \u00b7 5 \u00b7 5.<\/p>\n

    \n

    Question<\/h4>\n
    \n

    Which of the following shows the correct prime factorization of 120?<\/p>\n

      \n
    1. 2 \u00b7 60<\/li>\n
    2. 12 \u00b7 10<\/li>\n
    3. 2 \u00b7 6 \u00b7 10<\/li>\n
    4. 2 \u00b7 2 \u00b7 2 \u00b7 3 \u00b7 5<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      The correct answer is D. All answer choices are factorizations of 120, but choice D is the only prime
      \nfactorization. Each of the numbers is divisible only by 1 and itself.<\/div>\n<\/section>\n

      What is the LCM?<\/h3>\n

      The LCM<\/abbr> stands for the least common multiple<\/strong> of a set of numbers. It is the smallest\u00a0number that is a multiple of all numbers in a set.<\/p>\n

      How do I find the LCM?<\/h3>\n

      To find the LCM, first find the prime factorization of each\u00a0number in the set. Then look for the greatest number of times each\u00a0factor appears in any of the factorizations. Multiply these\u00a0together to find the LCM.<\/p>\n

      Find the LCM of 4, 6, and 18.<\/p>\n

      \"4<\/center>The LCM of 4, 6, and 18 is 2 \u00b7 2 \u00b7 3 \u00b7 3 =\u00a036, since 2 appears twice in the factorization of 4 and three\u00a0appears twice in the factorization of 18.<\/p>\n
      \n

      Question<\/h4>\n
      \n

      Which number is the LCM of 12, 21, and 24?<\/p>\n

        \n
      1. 3<\/li>\n
      2. 168<\/li>\n
      3. 6,048<\/li>\n
      4. 24<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is B. First, find the prime factorization\u00a0of each number in the set.<\/p>\n

        <\/p>\n

        Then look for the greatest number of times each factor appears\u00a0in any of the factorizations.\u00a0The LCM is 2 \u00b7 2 \u00b7\u00a02 \u00b7 3 \u00b7 7 = 168.<\/p>\n<\/div>\n<\/section>\n

        What is the GCF?<\/h3>\n

        The GCF<\/abbr><\/span> stands for the greatest common factor<\/strong> of a set of numbers. It is the largest\u00a0number that is a factor of all numbers in a set.<\/p>\n

        How do I find the GCF?<\/h3>\n

        To find the GCF, first find the prime factorization of each\u00a0number in the set. Then look for all numbers that appear in all\u00a0lists. Multiply these numbers to find the GCF. If there are no\u00a0common prime factors, the GCF of the set is 1.<\/p>\n

        Find the GCF of 30, 50, and 70.<\/p>\n

        <\/center>Both 2 and 5 appear in all three factorizations. Multiply them\u00a0together and find that the GCF of 30, 50, and 70 is 2 \u00b7 5 = 10.<\/p>\n
        \n

        Question<\/h4>\n
        \n

        What is the GCF of 18, 30, and 42?<\/p>\n

          \n
        1. 6<\/li>\n
        2. 18<\/li>\n
        3. 210<\/li>\n
        4. 630<\/li>\n<\/ol>\n<\/div>\n

          Reveal Answer <\/a><\/p>\n

          \n

          The correct answer is A. First, find the prime factorization\u00a0of each number.<\/p>\n

          <\/p>\n

          Since both 2 and 3 appear in all three factorizations, the GCF\u00a0is 2 \u00b7 3 = 6.<\/p>\n<\/div>\n<\/section>\n

          \n

          Important Tidbit<\/h4>\n

          Remember that the LCM will be no smaller than the largest number\u00a0in the\u00a0set. The GCF will be no bigger than the smallest number in the set.<\/p>\n<\/div>\n<\/section>\n

          <\/p>\n

          \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

          Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

          \u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Fundamental Theorem of Arithmetic Objective In this lesson, you will study and apply the Fundamental Theorem of Arithmetic and theoretic number concepts of\u00a0primes, factors, and multiples. Previously Covered: A prime number is a number divisible only by 1 and itself. A composite number is a number with factors other […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-8","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/8","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=8"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/8\/revisions"}],"predecessor-version":[{"id":361,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/8\/revisions\/361"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=8"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}