How To Find The Greatest Common Factor Of Two Numbers

Finding the greatest common factor, or GCF, of two numbers is useful in many situations in math, but particularly when it comes to simplifying fractions. If you're struggling with this or finding common denominators, learning two methods for finding common factors will help you achieve what you're setting out to do. First, though, it's a good idea to learn about the basics of factors; then, you can look at two approaches for finding common factors. Finally, you can look at how to apply your knowledge to simplify a fraction.

Factors are the numbers you multiply together to produce another number. For example, 2 and 3 are factors of 6, because 2 × 3 = 6. Similarly, 3 and 3 are factors of 9, because 3 × 3 = 9. As you may know, prime numbers are numbers that have no factors other than themselves and 1. So 3 is a prime number, because the only two whole numbers (integers) that can multiply together to give 3 as an answer are 3 and 1. In the same way, 7 is a prime number, and so is 13.

Because of this, it's often helpful to break down a number into "prime factors." This means finding all of the prime number factors of another number. It basically breaks the number down into its fundamental "building blocks," which is a useful step towards finding the greatest common factor of two numbers and is also invaluable when it comes to simplifying square roots.

The simplest method for finding the greatest common factor of two numbers is to simply list all of the factors of each number and look for the highest number that both of them share. Imagine that you want to find the highest common factor of 45 and 60. First, look at the different numbers you can multiply together to produce 45.

The easiest way to start is with the two you know will work, even for a prime number. In this case, we know 1 × 45 = 45, so we know 1 and 45 are factors of 45. These are the first and last factors of 45, so you can just fill in from there. Next, work out whether 2 is a factor. This is easy, because any even number will be divisible by 2, and any odd number won't. So we know that 2 isn't a factor of 45. What about 3? You should be able to spot that 3 is a factor of 45, because 3 × 15 = 45 (you can always build on what you know to work this out, for example, you'll know that 3 × 12 = 36, and adding threes to this leads you to 45).

Next, is 4 a factor of 45? No – you know 11 × 4 = 44, so it can't be! Next, what about 5? This is another easy one, because any number ending in 0 or 5 is divisible by 5. And with this, you can easily spot that 5 × 9 = 45. But 6 is no good because 7 × 6 = 42 and 8 × 6 = 48. From this you can also see that 7 and 8 aren't factors of 45. We already know 9 is, and it's easy to see that 10 and 11 aren't factors. Continue this process, and you'll spot that 15 is a factor, but nothing else is.

So the factors of 45 are: 1, 3, 5, 9, 15 and 45.

For 60, you run through the exact same process. This time the number is even (so you know 2 is a factor) and divisible by 10 (so 5 and 10 are both factors), which makes things a bit easier. After going through the process again, you should see that the factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Comparing the two lists shows that 15 is the greatest common factor of 45 and 60. This method can be time consuming, but it's simple and it will always work. You can also start at any high common factor you can spot straight away, and then simply look for higher factors of each number.

The second method of finding the GCF for two numbers is to use prime factors. The process of prime factorization is a little easier and more structured than finding every factor. Let's go through the process for 42 and 63.

The process of prime factorization basically involves breaking the number down until you're only left with prime numbers. It's best to start with the smallest prime (two) and work from there. So for 42, it's easy to see that 2 × 21 = 42. Then work from 21: Is 2 a factor? No. Is 3? Yes! 3 × 7 = 21, and 3 and 7 are both prime numbers. This means the prime factors of 42 are 2, 3 and 7. The first "break" used 2 to get to 21, and the second broke this down into 3 and 7. You can check this by multiplying all of your factors together and checking you get the original number: 2 × 3 × 7 = 42.

For 63, 2 isn't a factor, but 3 is, because 3 × 21 = 63. Again, 21 breaks down into 3 and 7 – both prime – so you know the prime factors! Checking shows that 3 × 3 × 7 = 63, as required.

You find the highest common factor by looking at which prime factors the two numbers have in common. In this case, 42 has 2, 3 and 7, and 63 has 3, 3 and 7. They have 3 and 7 in common. To find the highest common factor, multiply all of the common prime factors together. In this case, 3 × 7 = 21, so 21 is the greatest common factor of 42 and 63.

The previous example can be solved more quickly this way too. Because 45 is divisible by three (3 × 15 = 45), and 15 is also divisible by three (3 × 5 = 15), the prime factors of 45 are 3, 3 and 5. For 60, it's divisible by two (2 × 30 = 60), 30 is divisible by two as well (2 × 15 = 30), and then you're left with 15, which we know has three and five as prime factors, leaving 2, 2, 3 and 5. Comparing the two lists, three and five are the common prime factors, so the greatest common factor is 3 × 5 = 15.

In the event that there are three or more common prime factors, you multiply them all together in the same way to find the greatest common factor.

If you're presented with a fraction like 32/96, it can make any calculations that come after it very complicated unless you can spot a way to simplify the fraction. Finding the lowest common factor of 32 and 96 will tell you the number to divide both by, to get a simpler fraction. In this case:

\(32 = 2 × 16\)
\(16 = 2 × 2 × 2 × 2\)
\(\text{So } 32 = 2^5 = 2 × 2 × 2 × 2 × 2\)

For 96, the process gives:

\(96 = 48 × 2\)
\(48 = 24 × 2\)
\(24 = 12 × 2\)
\(12 = 6 × 2\)
\(6 = 3 × 2\)
\(\text{So } 96 = 2^5 × 3 = 2 × 2 × 2 × 2 × 2 × 3\)

It should be clear that 2⁵ = 32 is the highest common factor. Dividing both parts of the fraction by 32 gives:

\(\frac{32}{96} = \frac{1}{3}\)

Finding common denominators is a similar process. Imagine that you had to add the fractions 15/45 and 40/60. We know from the first example that 15 is the highest common factor of 45 and 60, so we can immediately express them as 5/15 and 10/15. Since 3 × 5 = 15, and both numerators are also divisible by five, we can divide both parts of both fractions by five to get 1 /3 and 2/3. Now they are much easier to add and see that

\(\frac{15}{45} + \frac{40}{60} = 1\)