What is a Multiple?

Before we can find the Least Common Multiple, we need to make sure we understand what a multiple actually is.

Multiple

A multiple is the result of multiplying a number by any whole number (1, 2, 3, 4, …). For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, …

Think of multiples as the numbers you land on when you skip count by a given number. If you skip count by 5, you get: 5, 10, 15, 20, 25 … These are all multiples of 5.

Note

Every number has an infinite list of multiples — they go on forever! So the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, … without end.

Example

List the first six multiples of 7.

Multiply 7 by 1, 2, 3, 4, 5, and 6:

$7 \times 1 = 7$
$7 \times 2 = 14$
$7 \times 3 = 21$
$7 \times 4 = 28$
$7 \times 5 = 35$
$7 \times 6 = 42$

So the first six multiples of 7 are: 7, 14, 21, 28, 35, 42.

Common Multiples

When two or more numbers share a multiple, we call it a common multiple.

Common Multiple

A common multiple is a number that appears in the multiples list of two or more different numbers.

Analogy

Imagine two buses that leave the same stop. Bus A departs every 4 minutes and Bus B departs every 6 minutes. The times when both buses leave together are the common multiples of 4 and 6. These are the moments where their schedules line up!

Let's see this with numbers:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …

Multiples of 6: 6, 12, 18, 24, 30, 36, …

The numbers that appear in both lists are the common multiples: 12, 24, 36, …

Note

Just like multiples in general, there are infinitely many common multiples for any pair of numbers. That is why we focus on the lowest (or least) one.

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