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# Least Common Multiple

## Smallest number having the numbers as factors

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Least Common Multiple

Arjay is planning a barbecue. He is at the store to buy hot dogs and hot dog buns. The hot dogs come in packs of 8 while the buns come in packs of 6. At least how many packages of each should he buy to have the same number of hot dogs and buns?

In this concept, you will learn to find the least common multiples of numbers by using lists.

### Finding the Least Common Multiple Using Lists

Common multiples are multiples that two or more numbers have in common. The least common multiple (LCM) is the smallest multiple that two numbers have in common.

Let’s look back at the common multiples for 3 and 4.

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. The LCM of these two numbers is 12. It is the smallest number that they both have in common.

To find the LCM, list the multiples of the two numbers. Stop when you have found the first common multiple.

### Examples

#### Example 1

Earlier, you were given a problem about Arjay on his shopping trip.

Arjay wants to know at least how many packages of 8 hot dogs and 6 hot dog buns he needs to buy to get the same amount of each. Find the LCM of 8 and 6 to find the number of packages for each.

First, list the multiples of both 8 and 6. Stop at the first common multiple.

8 = 8, 16, 24, 32

6 = 6, 12, 18, 24

The LCM of 8 and 6 is 24.

Then, find number of packages of hot dogs and buns needed to make 24 hotdogs.

\begin{align*}24 \div 8 = 3\end{align*}

\begin{align*}24 \div 6 = 4\end{align*}

Arjay must buy 3 packages of hotdogs and 4 packages of hotdog buns to get the same number of each.

#### Example 2

Find the LCM of 20 and 15.

First, list out the multiples of both 20 and 15. Stop at the first common multiple.

20 = 20, 40, 60, 80, 100

15 = 30, 45, 60

The LCM Of 20 and 15 is 60.

#### Example 3

Find the LCM of the pair of numbers.

5 and 3

First, list the multiples of both 5 and 3. Stop at the first common multiple.

5 = 5, 10, 15, 20, 25

3 = 3 6, 9, 12, 15

The LCM of 5 and 3 is 15.

#### Example 4

Find the LCM of the pair of numbers.

2 and 6

First, list the multiples of both 2 and 6. Stop at the first common multiple.

2 = 2, 4, 6, 8

6 = 6

The LCM of 2 and 6 is 6.

#### Example 5

Find the LCM of the pair of numbers.

4 and 6

First, list the multiples of both 4 and 6. Stop at the first common multiple.

4 = 4, 8, 12, 16, 20

6 = 6, 12

The LCM of 4 and 6 is 12.

### Review

Find the LCM of each pair of numbers.

1. 3 and 5
2. 2 and 3
3. 3 and 4
4. 2 and 6
5. 3 and 9
6. 5 and 7
7. 4 and 12
8. 5 and 6
9. 10 and 12
10. 5 and 8

To see the Review answers, open this PDF file and look for section 5.10.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Common Multiple

A common multiple is a number that two or more given numbers can be multiplied to equal. For example: A common multiple of 5 and 7 is 35, since both 5 and 7 can be multiplied to equal 35.

Least Common Multiple

The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.

Multiple

A multiple of a number is the product of that number and a whole number.