Least common multiple

The least common multiple (LCM) of two or more numbers is the smallest whole number (except zero) that is a multiple of each number. There are several ways to find the least common multiple of two or more numbers.


listing multiples


Listing multiples is one way to find the LCM.

Example

To find the LCM of 2, 6, and 9, list the nonzero multiples of each number until you find one that is common to all three numbers.

2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
6: 6, 12, 18, ...
9: 9, 18, ...

Thus, the LCM of 2, 6, and 9 is 18. This is written as: LCM (2, 6, 9) = 18.


using prime factorization


Prime factorization can also be used to find the LCM.

Example

To find the LCM of 4 and 6, first write the prime factorization of each number.

4 = 2 × 2
6 = 2 × 3

Then multiply the prime factors of both numbers together. Include each prime factor as many times as it appears in 4 or 6. This Venn diagram will help you see how many times to include each factor:

2 × 2 × 3 = 12
LCM (4,6) = 12

See also least common denominator, common multiple.