 Another important notion in number theory is the notion of divisibility. So suppose I have three integers, p, m, and k, and their related p is equal to k times m. Now, I can say a number of different things at this point. First off, I can say that m divides p. I could also say that m is a divisor of p, or often we say this, p is divisible by m, and our notation really refers back to these first two phrasings. Not so much the third phrasing, but really the first two m divides p, m is divisor, and we have this notation m divides, there's a vertical bar, p. Now, since multiplication among the integers is commutative, whatever I can say about m, I can also say about k. So I could say k divides p, k is a divisor, p is divisible by k, k divides p. So all of these things are equally true about k and of m. Now, there's a couple of basic properties of divisibility that should be pretty obvious from the definition plus a little bit of algebra. What that means is that you should be able to prove them if asked. So that if I have m divides some number and m divides some other number, then m is going to divide the sum or difference of those two numbers. Likewise, if m divides a number, then m divides n times that number for any integer n. There are some things that also seem like they should be true, but are actually not going to be true in all cases. They will sometimes be false. Now, that doesn't mean they're always false. They are true in some cases. The problem is that because they're sometimes false, they're not reliable. We cannot rely on them for anything because we might be using one of the cases where they are false. So for example, one of the things that seems like it should be true, if m divides p and n divides p. So I have two different numbers that divide p, then the product m times n should divide p. And this seems like reasonable. This seems like something that should be true. But it's not. So 6 divides 12. 12 is, in fact, 6 times something. 4 divides 12. 12 is 4 times something. But the product 6 times 4, 24, does not, in fact, divide 12. So there's a counter example. Another one that seems like it should be true, if I have a number that doesn't divide p and a number that doesn't divide q, then that number can't divide p plus or minus q. So doesn't divide p, doesn't divide q, doesn't translate into doesn't divide there some or difference. And again, here's an example of that. 5 does not divide 8. 5 does not divide 12. And so 5 does not divide the sum 8 plus 12. Well, actually 8 plus 12 is 25 does, in fact, divide it. And one more thing that seems like it should be true, but it isn't. If m does not divide p, m does not divide q, then m does not divide the product p times q. Again, this is true in many, many, many cases, but it's false often enough that it is not reliable. 8 does not divide 12. 8 does not divide 2. But 8 does divide the product 12 times 2, the product of those two numbers. Again, the basic definition of divisibility can go a long way, especially if we keep in mind the fundamental theorem of arithmetic. So for example, I want to prove or disprove that some number divides some other number. And so what I'd like to do is to avoid having to compute what 5 to the 3rd times 7 to the 5th is, I'd like to avoid computing 140 to the 10th power. And most of all, I'd like to avoid having to do that division. So what can I do? Well, the fundamental theorem for arithmetic tells me that both of those numbers can be expressed as a unique product of primes. Now, again, 5 and 7 are both primes, so this number 5 to the 3rd, 7 to the 5th is already expressed as a product of primes. The only thing I have to worry about is that 140. And without going into the details as a product of primes, I could write 140 this way. So 140 to the 10th is with a little bit of algebra going to be this product. Now, what I'd like to, if possible, is to write this number as a product of this number and something else. And if I can, then this number divides. If it's not possible to do so, this number does not divide. So let's check that out. I'll rearrange those factors a little bit. So I need 3 5s and 5 7s. Well, I have 10 5s and 10 7s there, so I'll go ahead and separate those out and then I have what's left over. And so there's my 140 to the 10th. And since 140 to the 10th is now written as a product of this and something else, then 140 to the 10th is divisible by this and this number divides 140 to the 10th. So there's my conclusion and there is my proof of that conclusion. Similarly, I want to prove 112 divides 36 to the 15th or not. And again, we'll write all of our numbers as products of primes. 112 is 2 to the 4th times 7, 36, 2 squared times 3 squared. Raise that to the 15th is 2 to the 30th, 3 to the 30th, and let's see. So I'd like to write this number as a product of this number and something else. Well, there's a little bit of a problem here. This number requires a 7. If I want to write this number, I need a 7 and I don't have a 7 here. So that says it's impossible to write 36 to the 15th as a product of this and something else. And so that guarantees that 112 does not divide 36 to the power of 15.