A multiple is an integer that can be evenly divided by another integer. If c ⁄d is an integer, then c is a multiple of d. 45, 27, and 18, for example, are all multiples of 9. Alternatively, you could define a multiple as an integer with at least one factor. All that really matters is that you understand the concept of multiples, and this is best done with a simple example.

What are some multiples of 4?

How do we know these numbers are multiples of 4?

Also, note that any integer, n, is a multiple of 1 and n, because 1 n = n.

The least common multiple (LCM) of two integers is the smallest multiple that the two numbers have in common. Like the GCF, the least common multiple of two numbers is useful when manipulating fractions.

To find the LCM of two integers, you must first find the integers’ prime factorizations. The least common multiple is the smallest prime factorization that contains every prime number in each of the two original prime factorizations. If the same prime factor appears in the prime factorizations of both integers, multiply the factor by the greatest number of times it appears in the factorization of either number.

For example, what is the least common multiple of 4 and 6? We must first find their prime factorizations.

In this example, 2 appears as a prime factor of both integers. It appears twice in the prime factorization in which it is more prevalent, so to find the LCM we will use two 2s. We will also use the 3 from the prime factorization of 6. The LCM of 4 and 6 is therefore

Let’s try a harder example. What is the LCM of 14 and 38? Again, we start by finding the prime factorizations of both numbers:

In this example, 2 appears in both prime factorizations, but not more than once in each, so we only need to use one 2. Therefore, the LCM of 7 and 38 is

For practice, find the LCM of the following pairs of integers:

Compare your answers to the solutions: