Multiples can be shared by numbers. The lowest multiple shared by two numbers is called the **Lowest Common Multiple**, or **LCM**.

multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, ...

multiples of 12 are: 12, 24, 36, 48, 60, 72, ...

Multiples that appear in both lists are 36 and 72. The Lowest Common Multiple is 36, which is the first multiple that 9 and 12 share.

For larger numbers, it is easier to determine the LCM by using a Venn Diagram. Using a Prime Factor Tree, determine the product of the primes for each number.

Place these prime factors in a Venn Diagram. The union of the two sets represents the lowest common multiple. This can be written as for example) 9 ∪ 12, or 9 *union* 12.

The LCM (multiply each number on the Venn Diagram) = 2 x 2 x 3 x 3 = 36.

*Higher/Extended only*:To find the Lowest Common Multiple of more than two numbers, create a Venn Diagram with a set of prime factors for each number. The union of ALL the sets will give the factors for the Lowest Common Multiple for ALL the numbers.

What is the LCM of 8 and 10?

By inspection:

Factors of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

Factors of 10: 10, 20, 30, 40, 50, 60

The lowest common multiple is 40.

Answer: 40

In a game, one type of alien appears ever 22 seconds. A second type of alien appears every 36 seconds. If they have just appeared together, how long will it be until they appear together again?

This is a Lowest Common Multiple question. Using prime factors:

22: 2 x 11

36: 2 x 2 x 3 x 3

LCM = 2 x 2 x 3 x 3 x 11 = 396.

Check that 22 and 36 both divide into 396 without leaving a remainder.✔

Answer: 198 seconds

See also Venn Diagrams

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