 So an important concept when dealing with whole numbers is the notion of the greatest common divisor of a set of whole numbers. So let's start off with a few definitions. Remember, definitions are the whole of mathematics. All else is commentary. Definitions are very important. So first of all, we define a divisor as follows. Suppose n is equal to a times b, then a and b are divisors of n. Now, we actually have a different definition that's very similar. We also say the following. Suppose n equals a times b, then a and b are factors of n. Now if you look at these two definitions, you'll see there is a very important difference between the two of them. Here we call them divisors. Here we call them factors. In other words, factors and divisors are really the same thing just looked at from slightly different perspectives. Now you might object to the fact that we're calling the same thing two different names, but we do that with people too, so it's not that unusual. And here's a useful thing to keep in mind. The more important something is, the more ways we have to talk about it. And here, the fact that we have two ways of talking about a and b, we can either say they're divisors, or we can say they're factors, say that this relationship is actually a very important relationship. So let's say we want to find a divisor of 40. So remember, definitions are the whole of mathematics. All else is commentary. If I want to find a divisor, it helps to know what a divisor is. So we'll pull in that definition. And the idea here is that if I can write a number as a product, then the individual factors are going to be the divisors. So let's write 40 as a product, and we observe that maybe we can write 40 as 4 times 10. And so both 4 and 10 are divisors of 40. Because of the fundamental theorem of arithmetic, we know that every number can be written as a unique product of primes. We can then group the primes to form a factorization of the number. And factors are the same as divisors. So for example, let's find three different factorizations of the number 2 times 3 times 5 times 7. We'll do this the hard way first. First, we find 2 times 3 times 5 times 7. That's 210, and then we try to write 210 as a product. The difficulty here is that factoring is the hardest easy problem in mathematics. It's easy to explain what we're trying to do. We want to write 210 as a product. But it's actually very hard to find numbers that multiply to 210. So let's do this the easy way. We already have this number as a product. So if we group the numbers, we'll be able to form several factorizations. So maybe we'll keep the 2 and 3 together and the 5 and 7 together. And carrying out the multiplications that gives us 6 times 35 as one possible factorization. Or maybe we'll group the 2 and the 5 and the 3 and the 7. And so 10 times 21 is another factorization. And maybe we'll try 2, 7, and 3, 5. And so 14 times 15 is a third factorization. And this leads to the following idea. Given 2 or more numbers, the greatest common divisor, GCD, also called the greatest common factor, GCF, is the largest number that is a factor of all the given numbers. And again, we have 2 different terms for the same thing. And this is a reflection of how important the greatest common divisor or greatest common factor actually is. Remember, the more important something is, the more ways we have to talk about it. So let's find the GCD of 30 and 75. We'll find this the hard way first. So remember that when we're looking for divisors, we're looking for things that multiply to the number. So we want to find things that multiply to 30. So the numbers that multiply to 30, well, let's start with factor 1 times 30, gives us 30. And so we know that 1 and 30 are divisors. 2 times 15 equals 30, and so we know that 2 and 15 are divisors. 3 times 10 is 30, so 3 and 10 are divisors. 4 times, uh, nope, that doesn't work as a divisor. 5 times 6 is equal to 30, so 5 and 6 are divisors. And the next number we would try would be 6 times something, but we've already found that, which says that we've exhausted our list of possible divisors. And similarly, we can find the divisors of 75 by looking for the numbers that multiply to 75. And these are 1 times 75, 2 times, uh, nope, 3 times 25, 5 times 15, and there aren't any other numbers. And so our divisors are 75, 1, 3, 5, 15, 25, and 75. And after all this work, we've probably forgotten what our original problem is, so it's a good thing it's up here. We want to find the greatest common divisor and we stare at these lists of numbers and we see that the largest number that divides both is 5. Wait, no, it's actually 15. Now, that's a lot of work, so let's take a look at the easier way. Since any number can be expressed uniquely in terms of its prime factors, we find the prime factorization of 30 and of 75. And remember, the fundamental theorem of arithmetic says that the prime factorization is the recipe for a number. And so what we look for is the largest recipe that we can make in both of these factorizations. And here we see that both numbers have a 3 and both numbers have a 5, so this product 3 times 5 will be a factor of both numbers. And we can't include any other prime factors because if we include the 2, it won't be in the 75, and if we include the other 5, it won't be in the 30.