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Factors and Multiples

  • If a number divides another number exactly, without any remainder, the first number is a factor of the second one. 1 is a factor of all numbers.
  • prime number has only two factors; itself and 1.
  • There are an infinite number of prime numbers; the first ten are 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29.
  • The prime factors of a number are the factors of a number that are prime.
  • If we write 2 × 2 × 2 × 2 × 2 in the form 25, the 5 is called the index or power.
  • Indices is the plural of index.
  • The indices 2 and 3 have special names; square and cube respectively.
  • If two or more numbers all have the same factor, we call it a common factor. E.g. 5 is a common factor of 15, 20, and 30.
  • The largest number that is a common factor of two or more numbers is called their highest common factor or HCF. E.g. 8 is the HCF of 24, 40, and 56.
  • If two or more numbers all have a common factor, then the numbers are multiples of the common factor. E.g 5, 10, 15, 20 – are multiples of 5.
  • If a number is a multiple of two or more numbers, we say it is a common multiple. E.g. 20 is a common multiple of 5 and 2.
  • To get the common multiple of two or more numbers, multiply the numbers together, and you can multiply that result by any other number (1, 2, 34, 77, etc.) and the result would also be a multiple of the numbers.
  • The smallest number that is a common multiple of two or more numbers is called the lowest common multiple, or LCM. E.g 10 is the LCM of 5 and 2.
Examples
  1. Express 15288 as a product of its prime factors.
    • 15288 = 2 × 7644
    •            = 2 × 2 × 3822
    •            = 2 × 2 × 2 × 1911
    •            = 2 × 2 × 2 × 3 × 637
    •            = 2 × 2 × 2 × 3 × 7 × 91
    •            = 2 × 2 × 2 × 3 × 7 × 7 × 13
    • Thus 15288 = 2 × 2 × 2 × 3 × 7 × 7 × 13
  2. Express
    1. 9 × 9 × 9 × 9 in index form.
      • There are four 9’s multiplied together;
      • ∴ 9 × 9 × 9 × 9 in index form is 94
    2. 675 as a product of its primes in their index form.
      • 675 = 3 × 3 × 3 × 5 × 5
      • ∴ 675 = 33 × 52
  3.  Find the HCF of 18, 24, and 42.
    • 18 = 2 × 3 × 3
    • 24 = 2 × 2 × 3
    • 42 = 2 × 3 × 7
    • The common prime factors are 2 and 3.
    • ∴ The HCF = 2 × 3 = 6
  4. Give three common multiples of
    1. 3 and 11
      • 33 (3 × 11) is a common multiple of 3 and 11. Likewise 66 (33 × 2) and 99 (33 × 3).
    2. 4, 5, and 6
      • 120 (4 × 5 × 6) is a common multiple of 4, 5, and 6. Likewise 240 (120 × 2) and 360 (120 × 3)
  5. Find the LCM of 24, 28, 36, and 50.
    • In indices: 24 = 23 × 3
    •                      28 = 22 × 7
    •                      36 = 22 × 32
    •                      50 = 2 × 52
    • The prime factors of 24, 28, 36, and 50 are 2, 3, 5, and 7.
    • The prime factors of 24, 28, 36, and 50 with their highest indices are 23, 32, 52, and 7 (71).
    • ∴ The LCM of 24, 28, 36, and 50 is 23 × 32 × 52 × 7 = 12600
  6. Find the HCF of 216, 288, and 360.
    • The circled numbers are the common numbers in 216, 288, and 360.
    • ∴ The HCF = 2 × 2 × 2 × 3 × 3 = 72
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