# 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