E. The LCM - Least Common Multiple

Least common multiples are used regularly to simplify work with fractions, or to determine when two events might occur simultaneously.

Least Common Multiple
the smallest identical multiple shared between two or more numbers

Example 1

Find the least common multiple of 6 and 9.

List the multiples of 6 and 9 until you find a common multiple.

Multiples of 6 Multiples of 9
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
6 × 5 = 30
6 × 6 = 36		 9 × 1 = 9
9 × 2 = 18
9 × 3 = 27
9 × 4 = 36
9 × 5 = 45
9 × 6 = 54
6,12, 18,24,30,36, ... 9, 18,27,36,45,54, ...

Although 6 and 9 have several common multiples, the least common multiple is 18.


Could prime factorization help you determine the LCM of two numbers? Try the same example again using prime factorization.

 

Example 2

Using the prime factorization method, find the least common multiple of 6 and 9.

START List the multiples of 6 and 9 until you find a common multiple.

Prime Factorization of 6 Prime Factorization of 9

Identify the common factor and use it just once. In this example, there is a common factor of 3.

Then, identify the factors that are unique to the other numbers. In this example, the factor of 2 is unique to 6 and the additional factor of 3 is unique to 9. The product of 2, 3, and 3 is the LCM of 18. END



Using prime factorization is particularly helpful when finding the LCM of large numbers.

The following video demonstrates how to use prime factorization to find the LCM of larger numbers where listing multiples isn't practical: