Have you ever wondered, “What’s so great about the greatest common factor?” Or maybe you’re wondering what that even means. Well, first let’s review what a factor is. Then we’ll discuss what it means to have common factors. And finally, we’ll get to that whole thing about being “the greatest”. A factor is a whole number that’s part of a multiplication problem. So if you ask, “What factors does a number have?”, you’re asking, “What numbers could you multiply together to get that number?” For example, if you multiply 3 and 4 together you get 12. That means both 3 and 4 are factors of 12. Or if you multiply 6 and 7 together you get 42. So 6 and 7 are factors of 42. But it’s important to realize that numbers can have more than two factors. Take 12 for example. We multiplied 3 times 4 together to get 12, but we also could have gotten 12 by multiplying 2 times 6. That means 2, 3, 4 and 6 are all factors of 12. Can you think of any other whole numbers you can multiply together to get 12? Well, there’s 1 x 12 of course. But that’s kinda obvious because you can multiply any number by 1 and get back that number itself. That means that 1 is always a factor of any number, and the number itself is also always one of its own factors. Those factors (1 and the number itself) might seem just a little too easy to find which is why they’re called “trivial factors”. So now we have a list of all of the possible factors of 12… {1, 2, 3, 4, 6 and 12}. Those are the only whole numbers that can be multiplied together to get 12. But what does it mean for factors to be “common” factors? Well, if two friends like playing the same sport, you’d say they have a common interest, right? Similarly, if two numbers have the same factor, they have a common factor. To actually see what a common factor is, let’s get another number and figure out what its factors are. Let’s do that with 42 since we already know that 6 and 7 are factors. What other whole numbers can we multiply to get 42? Well, there’s 2 times 21. And 3 times 14 also gives you 42. Besides that, the only factors of 42 are its trivial factors of 1 and 42. So let’s put all of those factors into another factor list. Now, when we put these two factor lists next to each other, do you notice anything? Yep… there are several numbers that are common to both lists. Of course, 12 and 42 both share the factor 1, (since all numbers have a factor of 1) but they also share the factors 2, 3, and 6. That’s a lot of common factors! I’ll bet these numbers would be good friends, even though there are some factors that they don’t have in common. Ah, but now we can ask, “What’s their greatest common factor?” Yep, that would be the number 6 because it’s the common factor that has the largest value. Greatest basically just means the largest, so you can think of it as the largest common factor if that helps you remember better. So, to find the greatest common factor of any two numbers, all you have to do is list all of their factors and pick the one that has the largest value that’s in both lists. It’s as simple as that! But, why would you ever want to do that anyway? Well, one very common use for the greatest common factor is for simplifying a fraction. A fraction is inherently made up of two numbers. If the numerator and denominator have a common factor, that means that the fraction is not as simple as it could be because it contains factors that could have been canceled. For example, if you have the fraction 12 over 42, you know it can be simplified because 12 and 42 have common factors. The way most people would probably try to simplify 12 over 42 is by noticing that they are both even numbers which means they have a common factor of 2. So you could divide both the top and bottom by 2 to get the new, equivalent fraction 6 over 21. That is simpler than 12 over 42, but it's not as simple as the fraction could be because 6 and 21 still share a common factor of 3 that can be canceled. But, what if right at the beginning of the problem you knew that the greatest common factor that these numbers shared was 6? Then, if you simplified it by dividing both the top and the bottom of the fraction by 6, you’d get the result 2 over 7. And 2 over 7 is the simplest form this fraction can take. So knowing the greatest common factor can help you simplify a fraction in basically just one step. Of course, if it takes a lot more steps to figure out what the greatest common factor is, it might not save you as much time as you’d like. Okay, I hope that helps you understand what the greatest common factor is. Keep practicing and be sure to check out www.mathantics.com for help with other math topics.