How to Calculate the Least Common Multiple

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The least common multiple (LCM) of two or more numbers is used to determine the least common denominator (LCD) when adding fractions with unlike denominators. Use prime factorization to find the LCM and convert unlike denominators before adding.

Least Common Multiple (LCM) Definition

The term ​common multiple​ refers to a number that is a multiple of a set of at least two numbers. For example, the number 12 is a common multiple of 2 and 3 since it can be evenly divided by both numbers with no remainder.

2 × 6 = 12 \\ 3 × 4 = 12

The ​least common multiple​ (LCM) is the smallest number that can be evenly divided by all numbers in a set. Zero is not considered. For 2 and 3, 12 is a common multiple, but 6 is the least common multiple.

2 × 3 = 6 \\ 3 × 2 = 6

A set of numbers can have several common multiples but only a single least common multiple.

Using LCM to Find an LCD

The LCM of two or more numbers can be used when you are trying to add fractions with unlike denominators, such as 1/4 and 1/3. Adding fractions in this form requires you to find a ​common denominator,​ and rewrite each fraction to use that denominator before adding. If you first find the LCM of the unlike denominators, you can use it as the ​least common denominator​ (LCD). Rewriting each fraction using the LDC means you won’t have to simplify the result.

Finding a Least Common Multiple

There are a few different ways to find the LCM of two or more numbers. One of the simplest is to list all the multiples of each number and then determine the lowest number that appears in all lists. For 1/4 and 1/3, some of the multiples of 4 are {4, 8, 12, 16, 20}. For 3, multiples are {3, 6, 9, 12, 15}. Comparing these two sets, you can see that the smallest number appearing in each set is 12.

Prime factorization​ is another way to find the LCM. Instead of listing the multiples of each number, write its prime factorization. You then create a list that includes each unique factor the greatest number of times it appears in either factorization. Multiply the numbers in the list and you have the LCM. The following example shows how prime factorization works for the numbers 12 and 18.

Find the prime factorization for each number:

12 = 2 × 2 × 3 \\ 18 = 2 × 3 × 3

List each factor. For 2, use the factorization from the number 12 since 2 appears twice in that factorization. For 3, use the factorization from 18. Multiply the list of factors for the LCM.

2 × 2 × 3 × 3 = 36

The least common multiple of 12 and 18 is 36.

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