Lowest Common Multiple

Lowest Common Multiple

If m and n are integers, the lowest multiple shared by m and n is the lowest strictly positive integer that is a multiple of both m and n.

Notation

The notation for the “lowest common multiple” of several numbers is “LCM(a, b, …, z)” which means “the lowest common multiple of a, b, …, z”.

Properties

  • The LCM is always a positive integer.
  • The relationship between the GCD and the LCM.
    Consider: GCD(m, n) = p and PPCM(m, n) = q.
    Then: GCD(m, n) × PPCM(m, n) = m × n.
    And we can write: p × q = m × n.
    If GCD(8, 12) = 4 and PPCM(8, 12) = 24, then: 4 × 24 = 8 × 12.
  • By extension, we can find the LCM of two or more polynomials.
    x\(^{2}\) – 9 = (x + 3)(x – 3)
    x\(^{2}\) – x – 12 = (x + 3)(x – 4)
    x\(^{2}\) + 6x + 9 = (x + 3)(x + 3)
    Therefore: the LCM of these three polynomials is: (x + 3)(x + 3)(x – 3)(x – 4).

Examples

  • If mult(12) = {0, 12, 24, 36, 48, 60, 72, 84, …} and mult(15) = {0, 15, 30, 45, 60, 75, 90, …},
    then: LCM(12, 15) = 60.

    • We can see that: 12 = 2 × 2 × 3 and 15 = 3 × 5
    • Therefore: LCM(12, 15) = 2 × 2 × 3 × 5 = 60
  • If mult(9) = {0, 9, 18, 27, 36, 45, 54, 63, 72, …} and mult(21) = {0, 21, 42, 63, 84, …}, then:
    LCM(9, 21) = 63.

    • We can see that: 9 = 3 × 3 and 21 = 3 × 7
    • Therefore: LCM(9, 21) = 3 × 3 × 7 = 63

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