1.04 Multiples and least common multiple (LCM)

A multiple is the result of multiplying a number by an integer.

Remember, some numbers share multiples. For example, 12 is a common multiple of 2 and 3 because 2\times6=12 and 3\times4=12.

When we are asked to find the least common multiple, we are being asked to find the smallest multiple that is shared by two numbers.

Let's say we find the LCM of 6 and 8.

We can list the multiples of 6 and 8 as follows:

• 6:\, 6, 12, 18, 24, 30, 36
• 8:\, 8, 16, 24, 32, 40

Using a Venn diagram, we can see the multiples of 6 and 8 listed in each circle as two sets. The intersecting part represents the similarity or intersection between the two sets.

The Venn diagram shows that the least common multiple of 6 and 8 is 24.

Examples

Example 1

What is the least common multiple of 7 and 4?

Worked Solution

Create a strategy

We can consider the first seven multiples of 7 and the first seven multiples of 4, and choose the least number that appears in both lists.

Apply the idea

The multiples of 7 are: 7,\,14,\,21,\,28,\,35,\,42,\,49

The multiples of 4 are:4,\,8,\,12,\,16,\,20,\,24,\,28

The least number in both list is the LCM, 28.

Watch question walkthrough

Example 2

Use the following Venn diagram to find the LCM of 2,\,3, and 4

Worked Solution

Create a strategy

When we are asked to find the least common multiple, we are being asked to find the smallest multiple that is shared by two or more numbers.

Apply the idea

The Venn diagram shows the three subsets of the mulitiples of 2, 3 and 4. The two multiples common to all three numbers are 12 and 24. The least common multiple, therefore is 12.