How do you find that the LCM of 72 and 108 is 216?

Answer 1

To find the Least Common Multiple, you factor both values, then multiply the highest exponent of each factor. (You are removing common factors to find a multiple that is smaller than just multiplying the two numbers together.)

Example: for 72 and 108, the obvious multiple is 72 x 108 = 7776, but we can find a smaller one.

72 = 2 x 2 x 2 x 3 x 3 --- (23 x 32)

108 = 2 x 2 x 3 x 3 x 3 --- (22 x 33)

The highest exponent of each factor is 23 and 33, and 8 x 27 = 216

You can see that 216 is (72 x 3) or (108 x 2). There is no smaller even multiple.

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Rather than using the exponential formula, you can also multiply the common factors by the uncommon factors.

Common factors = 2 x 2 x 3 x 3 (36)

Uncommon factors = 2 and 3 (6) and 36 x 6 = 216