Least Common Multiple

Lesson 3-6

Multiples A multiple is formed by multiplying a

given number by the counting numbers. The counting numbers are 1, 2, 3, 4, 5,

6, etc.

Example: List the multiples of 4:

4 x 1 = 4 4 x 2 = 8 4 x 3 = 12 4 x 4 = 16 4 x 5 = 20 4 x 6 = 24

Counting Numbers

So, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, etc.

What are the first five multiples of 13?

13 x 1 =13

13 x 2 = 26

13 x 3 = 39

13 x 4 = 52

13 x 5 = 65

13, 26, 39, 52, 65

Find the Missing Multiples

6, 12, 18, ____, ____

___, 6, 9, 12, ____, ____, 21

___, 24, 36, 48, 60, ____

24 30

3 15 18

12 72

Least Common Multiple (LCM)

The least common multiple is the smallest number that is common between two lists of multiples.

EXAMPLE: Find the LCM of 12 and 18

The multiples of 12:

•12 x 1 = 12

•12 x 2 =24

•12 x 3 = 36

•12 x 4 = 48

•12 x 5 =60

The multiples of 18:

•18 x 1 = 18

•18 x 2 = 36

•18 x 3 = 54

•18 x 4 = 72

•18 x 5 = 90

12, 24, 36, 48, 60

18, 36, 54, 72, 90

The first number you see in both lists is 36.

The least common multiple of 12 and 18 is 36.

Example 2: Find the LCM of 9 and 10

9, 18, 27, 36, 45, 54, 63, 72

10, 20, 30, 40, 50, 60, 70, 80If you don’t see a common multiple, make each list go further.

81, 90, 99

90, 100, 110

The LCM of 9 and 10 is 90

Example 3:Find the LCM of 4 and 12

4, 8, 12, 16

12, 24, 36Answer: 12

Using Prime Factorization to find the LCM

• Make factor trees for each number. Write the prime factorization in exponent form.

•Identify all of the prime numbers among the prime factorizations. Every prime number will be used.

• When the same prime number occurs in more than one prime factorization, select the prime that there’s more of (for example, 25 beats 22.)

• Multiply your prime numbers to get the LCM.

Example: Find the LCM of 88 and 102

88

2 x 44

2 x 22

2 x 11

102

2 x 51

3 x 17

88 = 23 x 11102 = 2 x 3 x 17

We have to use every prime number. The numbers we will use are 2, 3, 11, and 17.

We have 2 in both places. Select the “bigger” one. 23 beats regular 2.

LCM: 23 x 3 x 11 x 17 = 4,488

Homework Time