Charlie is playing a video game. He has to grab the magic token when the guard turns his back. He notices that the magic token appears every 4 seconds and the guard turns his back every 3 seconds. How can Charlie use this information to get the timing right so he can grab the magic token?

In this concept, you will learn to identify multiples and find common multiples for pairs of numbers.

### Finding Common Multiples

A **multiple** is the product of a quantity and a whole number. Here are some multiples for the quantity of 3, multiplied by different whole numbers.

Listing out these products is the same as listing out multiples.

3, 6, 9, 12, 15, 18...

You can see that this is also the same as counting by threes. The dots at the end mean that these multiples can go on and on and on. Every number has an **infinite** number of multiples.

List six multiples for 4.

To do this, think of taking the quantity 4 and multiplying it by 1, 2, 3, 4, 5...

Our answer is 4, 8, 12, 16, 20, 24...

A **common multiple** is a multiple that two or more numbers have in common. List the multiples of the numbers to find the common multiples.

Find common multiples for 3 and 4.

First, write out the first few multiples for the numbers and then identify the multiples the two numbers have in common.

3, 6, 9, **12**, 15, 18, 21, **24**, 27, 30, 33, **36**

4, 8, **12**, 16, 20, **24**, 28, 32, **36**, 40, 44, 48

Some of the common multiples of 3 and 4 are 12, 24, and 36.

### Examples

#### Example 1

Earlier, you were given a problem about Charlie and his video game.

Charlie has to grab the magic token that appears every 4 seconds when the guard turns his back, which is every 3 seconds. Find the common multiples of 3 and 4 to find out when both will occur at the same time.

First, list the first few multiples of 3 and 4.

3, 6, 9, 12, 15, 18, 21, 24, 27

4, 8, 12, 16, 20, 24, 28, 32, 36

Then, identify the common multiples.

3, 6, 9, **12**, 15, 18, 21, **24**, 27

4, 8, **12**, 16, 20, **24**, 28, 32, 36

Some common multiples of 3 and 4 are 12 and 24. Charlie can grab the magic token 12 or 24 seconds after he enter that stage. Note that 12 and 24 are multiples of 12. Charlie will get a chance to grab the magic token every 12 seconds.

#### Example 2

What are common multiples of 3 and 7?

First, write out the first few multiples for the numbers.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Then, identify the multiples the two numbers have in common.

3, 6, 9, 12, 15, 18, **21**, 24, 27, 30

7, 14, **21**, 28, 35, 42, 49, 56, 63, 70

A common multiple of 3 and 7 is 21.

In this list, the only common multiple between 3 and 7 is 21. You could find more multiples by increasing the length of the list of multiples for 3 and 7 or finding a multiple of the common multiple.

42 is a multiple of both 3 and 7.

#### Example 3

List eight multiples of 6.

First, multiply 6 by 1 through 8.

The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, and 48.

#### Example 4

List six multiples of 8.

First, multiply 8 by 1 through 6.

Six multiples of 8 are 8, 16, 24, 32, 40, and 48.

#### Example 5

What are the first two common multiples of 6 and 8?

First, list the first few multiples of 6 and 8.

6, 12, 18, 24, 30, 36, 42, 48 ...

8, 16, 24, 32, 40, 48, ...

Then, identify the common multiples.

6, 12, 18, **24**, 30, 36, 42, **48** ...

8, 16, **24**, 32, 40, **48** ...

Two common multiples of 6 and 8 are 24 and 48.

### Review

List the first five multiples for each of the following numbers.

- 3
- 5
- 6
- 7
- 8

Find two common multiples of each pair of numbers.

- 3 and 5
- 2 and 3
- 3 and 4
- 2 and 6
- 3 and 9
- 5 and 7
- 4 and 12
- 5 and 6
- 10 and 12
- 5 and 8

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.9.