Least Common Multiple

The Least Common Multiple (LCM) of two numbers, a and b, is the smallest number that has both a and b as factors.

Suppose that you really, really, really want to find the LCM of 12 and 15, or else your mom won't let you go to the movies. The most reliable way to find the LCM of two numbers is by first finding the prime factorizations of the two numbers. For 12 and 15, they would be:
12 = 2² * 3
15 = 3 * 5.

Now, the prime factorization of the LCM will be just big enough to include both 2² * 3 and 3 * 5. Therefore, the LCM is equal to:
2² * 3 * 5
= 60

Notice that you only have to include the 3 once, even though 3 is part of the prime factorization for both 12 and 15. It's okay (in fact, it's necessary) for a factor to serve double-duty like that if it shows up in both prime factorizations.

Here's another example. You're looking for the LCM of 20 and 24. So you dutifully write out the prime factorization of both numbers:
20 = 2² * 5
24 = 2³ * 3

Like in the previous example, you know that the 2s will have to serve double-duty. But what is the exponent on the 2? Are there two 2s, or are there three 2s?

The answer is three. 2² is included as a factor of 2³, but 2³ is too big to be a factor 2². So, remember: in cases like this, choose the bigger exponent.

The LCM of 20 and 24, therefore, is:
2³ * 3 * 5
= 120.

When will you use the LCM? Well, here are just a few uses:

When you're finding the lowest common denominator for adding or subtracting two fractions, you're really finding the LCM of the two denominators.
If you know that a and b are factors of some number N, then you know that the LCM of a and b is also a factor of N. For example, if you know that a number has 12 and 15 as factors, then this number must be a multiple of 60.