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*\(^{2}\) + 6*x*+ 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