Shortcuts to Least Common Multiple and Greatest Common Factor

These shortcuts will help you find the Least Common Multiple and the Greatest Common Factor of any set of numbers.

Least Common Multiple:

Find the least common multiple for 168 and 40:

1. Do a factor tree on each number.

168 40

2 x 84 2 x 20

2 x 2 x 42 2 x 2 x 10

2 x 2 x 6 x 7 2 x 2 x 2 x 5

2 x 2 x 2 x 3 x 7

2. Match up all of the same numbers in the prime factorization. (The final row of numbers is your prime factorization) Any numbers that do not match you will leave off to the side. Example:

168 = 2 x 2 x 2 x 3 x 7

40 = 2 x 2 x 2 x 5

Since all of the twos match up, you use that combination only once. In essence, one set of twos cancels out the other. The three, seven and five do not match up, so you simply multiply them with the set of 3 twos you have.

2 x 2 x 2 x 3 x 7 x 5 = 840

840 is your Least Common Multiple (LCM)

Greatest Common Factor:

Please remember the first step before completing any of the steps below is to determine whether the smaller number can divide evenly into the larger number. If it is able to be divided evenly, then you have found your greatest common factor! There is no need to go any further.

Find the Greatest Common Factor for 168 and 40:

1. Do a factor tree on each number.

168 40

2 x 84 2 x 20

2 x 2 x 42 2 x 2 x 10

2 x 2 x 6 x 7 2 x 2 x 2 x 5

2 x 2 x 2 x 3 x 7

2. Circle ONLY the matching numbers in each prime factorization. (The final row of numbers is your prime factorization) Example:

168 = 2 x 2 x 2 x 3 x 7

40 = 2 x 2 x 2 x 5

3. Multiply ONLY the numbers that match up.

2 x 2 x 2 = 8

8 is your Greatest Common Factor (GCF)

LCM and GCF EXAMPLE for 300 and 36:

300 36

2 x 150 4 x 9

2 x 2 x 75 2 x 2 x 3 x 3

2 x 2 x 3 x 25

2 x 2 x 3 x 5 x 5

2 x 2 x 3 x 5 x 5

2 x 2 x 3 x 3

LCM = 2 x 2 x 3 x 3 x 5 x 5 = 900

2 x 2 x 3 x 5 x 5

2 x 2 x 3 x 3

GCF = 2 x 2 x 3 = 12

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