Prime factorization as a tool for GCF and LCM

I’ve actually blogged about this before, but I think it’s both interesting and useful to see it all in one place.

Prime factorization is probably the quickest, most foolproof way to determine the GCF or LCM for two numbers. In fact, prime factorization gives you everything you need to find both the GCF and the LCM in a single bit of work.

Let’s say we’re asked to find the GCF for 12 and 16. We’ll start by finding their prime factors.

12: 2 * 2 * 3

16: 2 * 2 * 2 * 2

To find the GCF, we look for common prime factors.

12: 2 * 2 * 3

16: 2 * 2 * 2 * 2

Both lines have 2 *2 bolded, so I multiply them together and learn that the GCF for 12 and 16 is 4.

But what if we’re suddenly asked to find the LCM of both numbers instead. Fortunately, we just found the prime factors for 12 and 16 for our GCF problem. Let’s look at them now to see how we can use them to solve for LCM.

12: 2 * 2 * 3

16: 2 * 2 * 2 * 2

We have the exact same prime factors, but how can we use that to find the LCM? Oddly enough, we do it by using the numbers left over after we found the GCF.

12: 2 * 2 * 3

16: 2 * 2 * 2 * 2

For 12, we were left with 3; and for 16, we were left with 2 * 2, or 4. That’s pretty interesting.

We now multiply 12 and 4 together to get 48, the LCM for 12 and 16. Just to be sure, though, we also multiply 16 and 3, and discover that it also equals 48.

If we have the prime factors for two numbers, we use the ones they have in common to find their GCF and the ones they don’t have in common to find LCM. We’ve just used the same information to cut our work in half.