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[latex]^{2}[/latex] – 9 = (x + 3)(x – 3) x[latex]^{2}[/latex] – x – 12 = (x + 3)(x – 4) x[latex]^{2}[/latex] + 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