 So, let's talk about divisors and divisibility, and here's a useful general rule of life. The more words we have for a concept, the more important the concept tends to be. So, for example, let's take 3 plus 5 equals 9, and we could say that this is an addition sentence. We could also call it a total. We might also call it a sum. It happens to be an incorrect addition sentence, total or sum, but nevertheless, it is still an addition sentence, a total, a sum, and we have lots and lots of words for this same basic thing, and so that suggests that this type of concept, this putting together that we call addition, is an important idea. And so, based on the number of ways we have of talking about it, the following concept is extremely important. So, as I have integers a, b, and c, and I know that c is equal to a times b, then I can speak about the relationship between them and the following ways. First of all, I might say that c is a multiple of b, c is some number of b's, it's a multiple. I could also say that b is a divisor of c. I could also shorten that somewhat and say that b divides c, and in this particular case when I say that I also have a way of writing that, that's b vertical bar, c. I might also say that c is divisible by b. Note the inversion of our clauses here, b divides c, that's the active form, passive form c is divisible by b. And we could also say that a and b both are factors of c. And finally, we could say that c is the product of a and b. And so that's one, two, three, four, five, six ways we can talk about the relationship. Actually, there's even more because commutativity still holds. If c is equal to a times b, then c is also equal to b times a. So that c is the multiple of a, a is divisor, a divides, c is divisible by a, and so on. So there's about, so that puts us up to what, four, eight, nine, ten different ways we can describe this relationship. And because this is a definition, it is absolutely essential that you learn it and understand all of its implications. You can't do mathematics if you don't know the definitions. You have to know the definitions. It is impossible to do mathematics without them. Or at least it can be very, very, very, very, very, very hard to do mathematics without the definitions. For example, here's a nice simple problem. Determine whether 338,195 times 5 times 1,895 times 28,951 times 3 is divisible by 15. And so here's the hardest way possible to do this problem. So first of all, maybe I'll multiply everything out because, well, it says times, I guess I'll multiply. And so I do this multiplication and a couple of weeks from now I come back with the correct answer. Well, wait, I'm not done with the problem yet. We did want to determine whether that's divisible by 15. So I will see if I can divide the one by the other. So I'll do the division this thing divided by 15. And a couple of weeks later I come back with the answer. Yes, it is in fact divisible by 15. And this is the hardest way possible to do this problem. It could be done this way, but let's see if we can make this method a little bit more efficient. And how we're going to do that is we're going to go back to the definition of what it means for something to divide something else. B divides C if I can write C as a product of something times B. So let's see if I can do that. So our definition of divisibility, if C is a product A times B, then C is divisible by B. So if I can write this thing as a product of something and 15, we'll be able to answer the question immediately. So let's see. Well, I might use the fact that I do actually know some multiplications. And here's an important thing to notice. I am not dividing here. Division is the hardest of the elementary operation. So if we have any chance of avoiding it, we'll take it. I want to see if I can find this as a product of something in 15. And I do happen to know that 15 is 3 times 5. And I have a 5 here. I have a 3 here. And the associative and commutative property of multiplication says I can rearrange them so that 3 and the 5 are together. And I can combine them. That's the associative property. So now I have that 15 there. And now I have C as a number times 15. And that's all I need. So as soon as I have this thing as a number times 15, I know that it's divisible by 15. Now, if you look closely, that really only worked because we had a 3 and a 5 among the factors. And we happen to know 3 times 5 is equal to 15. And what that suggests is this property that I can write a number as a product of smaller numbers is going to be important. So that suggests another definition, which is the following. A whole number greater than 1. And that's an important restriction. A whole number greater than 1 is composite. If I can write it as the product of smaller whole numbers, it's prime if it can't be. So 1 itself is not considered prime or composite. We'll discuss the reasons for that later. It has to do with the fundamental theorem of arithmetic. So let's take a look at that, determine whether the following numbers are prime or composite, then prove it. So our numbers are 12 and 3. And prove it means we have to go back to the definition of prime or composite. So let's see. Well, 12 is easy. I can write 12 as the product of smaller numbers. 12 is equal to 3 times 4. And that's all I need to determine that 12 is composite. So because I can write 12 as the product of smaller numbers, 12 is composite. How about 3? Well, I can try the same thing. I can try to write 3 as the product of smaller numbers. And well, there's only a few smaller numbers than 3. So 1 times 1, nope. 1 times 2, nope. 2 times 2, nope. And these are the only possible products of smaller numbers. Any other product will require larger numbers. So we have a product 2 times 3, 3 times 5. But then 3 is not smaller than 3. No product of smaller numbers will give us 3. And so that tells us that 3 is a prime number. Notice one important thing. It's really easy to prove a number is composite. It's really hard to prove a number is prime.