Least Common Multiples (LCMs)

A common multiple of two whole numbers a and b is a number c which a and b both divide into evenly.

For example, 48 is a common multiple of 6 and 12 since

48 ÷ 6 = 8   and

48 ÷ 12 = 4.

The least common multiple is just what it sounds like... the smallest of all the common multiples.

Example:

Find the least common multiple of 9 and 12.

To do this, we can list the multiples:

9: 9, 18, 27, 36, 45, 54, 63, 72...

12: 12, 24, 36, 48, 60, 72, ...

We see that 36 is the LCM.

Another way to find the LCM of two numbers is to divide their product by their greatest common factor (GCF).

In this example, the GCF of 9 and 12 is 3. The product of 9 and 12 is 108.

So, the LCM is 108 ÷ 3 = 36.

A third way to find the LCM of two numbers is to list all of the factors of each number and multiply all of the factors of either or both numbers.

Example:

9 = 3 · 3

12 = 2 · 2 · 3

LCM = 2 · 2 · 3 · 3 (Note: one of the 3’s is common to both numbers so it is listed multiplied only once.)

LCM = 36