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Finding the LCM of Expressions

• The LCD of the rational expressions is the same as the Least Common Multiple (LCM) of the two denominators.

• When adding and subtracting rational expressions without common denominators, the Lowest Common Denominator (LCD) must be found.

• The LCM of two integers a and b is the smallest positive integer that both a and b will divide into evenly.

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• Example 1

The smallest number that both 3 and 5 will divide into evenly is 15.

Find the LCM of 3 and 5.

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• Example 2

The smallest number that both 4 and 6 will divide into evenly is 12.

Find the LCM of 4 and 6.

Note that both 4 and 6 will divide evenly into 24, but 12 is the smallest such number, and is the LCM.

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• Example 3

This problem is more difficult and we will need another procedure rather than just working in our head.

Find the LCM of 36 and 45.

Write the prime factorization of each number. We will use factor trees to do this.

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36

4 9

2 2 3 3

2 236 2 3

45

5 9

5 3 3

245 3 5

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2 236 2 3 245 3 5

Use the prime factorizations to “build” the LCM.

Start with the smallest prime used in either factorization.

2The largest exponent on 2 in either factorization is …

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2 236 2 3 245 3 5

Use the prime factorizations to “build” the LCM.

Start with the smallest prime used in either factorization.

22The largest exponent on 2 in either factorization is …

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2 236 2 3 245 3 5

The next prime is 3.

22

The largest exponent on 3 in either factorization is …

3

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2 236 2 3 245 3 5

The next prime is 3.

22

The largest exponent on 3 in either factorization is …

23

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2 236 2 3 245 3 5

The last prime is 5.

22

The largest exponent on 5 in either factorization is 1.

23 5

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Multiply … 2 22 3 5

The LMC of 36 and 45 is 180.

4 9 5

• The same pattern used in this problem can be used with variable expressions.

180

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• Example 4

Find the LCM of the two expressions:

Write the prime factorization of each.

3

2

8

10

x

x

3 3

2

2

2 5

x

x

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3 3

2

2

2 5

x

x

32 5 3x 38 5 x 340x

Remember to use the largest exponent.

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• Example 5

Find the LCM of the two expressions:

Write the prime factorization of each.

2

22

12 2 4

40 4

x x x

x x

222 3 2 4x x x

23 22 5 4x x

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222 3 2 4x x x

23 22 5 4x x

32 3 2x 22x 2

4x 5

2 22120 2 4x x x

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