 One important concept that we want to discuss is that of divisibility, and so we have the following definition. Suppose I have P, M, and K, all integers, and suppose I know that P is equal to K times M. Then, I say one of several things. First of all, I say that M divides P. We also might say that M is a divisor of P, or we might say that P is divisible by M. Now, the last is actually the most common way we have of stating this particular property. We say 25 is divisible by 5, 100 is divisible by 4, and so on. However, for purposes of how we write this mathematically, this is actually the worst way to state this, because the mathematical notation we use is the following. If M divides P, we write M, and there's a vertical bar there, and then P. Now, there's some important properties about divisibility. These are basic properties. They should be fairly obvious, and you should be able to prove them by referring back to the definition of divisibility. So, for example, if M divides P and M divides Q, then M divides P plus or minus Q. Likewise, if M divides P, then M divides N times P for all integers N, and what's almost as important are identifying things that seem like they should be true, but in fact are not always true. They are sometimes false, and so they are not reliable. So, here's probably the most important one. Suppose I have M divides P, and then I have something else N also divides P. Then we'd like to think that the product MN will then divide P, but that's not true. So, here I have 6 divides 12, 4 also divides 12, but 6 times 4 is 24 is not a divisor of 12. Another property we think should be true, but is not always true, is that if M does not divide P and M does not divide Q, then M shouldn't divide the sum or difference of the two, and that's actually the inverse of this property, which actually is true, that if M divides something and M divides something else, M will divide the sum or difference. Now, if M doesn't divide, is it possible that it doesn't divide the sum or difference? Well, again, this is false. This is not something we can rely on. 5 does not divide 8. 5 does not divide 12, but if I look at 8 plus 12, that's 25 does in fact divide 20. And again, we have something that's similar to the inverse of this second property here. If M divides P, then M divides any multiple. Well, if M does not divide P, and likewise does not divide Q, we might suspect that M does not divide the product P times Q. And again, this is false. This is not true. We cannot reliably claim this. And as a good example of that, 8 does not divide 12, 8 does not divide 2, but if I take a look at 12 times 2, that's 24, 8 does in fact divide 24. So let's do a quick proof, prove or disprove. This number 5 to the 3rd, 7 to the 5th divides 140 raised to the 10th power. Now, one possibility is we could figure out what 140 divided to the 10th power is, and then see if 5 to the 3rd, 7 to the 5th actually divides it. But we should go back to our definition of divisibility. So to determine whether 5 to the 3rd, 7 to the 5th divides 140 to the 10th, what we'd like to do is we'd like to see if 140 to the 10th can be written as a product of 5 to the 3rd, 7 to the 5th, and something else. Now, the fundamental theorem of arithmetic says that we can express every number uniquely as a product of primes. And so what we should do is we should try and write both of these numbers in terms of a prime product. Now, 5 and 7 are primes, so this is the unique expression of whatever this number is in terms of its prime factors. 140 is not prime, so I can write that as a product of primes. And so that's going to give me that. Raise to the 10th power is that raised to the 10th power. And I can apply my standard rules of exponents. And what I'd like to do is I'd like to see if I can write this as 5 to the 3rd, 7 to the 5th times whatever's left over. And I can do that. I have a bunch of 5s here. I have a bunch of 7s here. And hopefully I will have enough, so I'll rearrange my factors a little bit. And there's my 140 raised to the 10th, 5 to the 3rd, 7 to the 5th times some other stuff. And so I know that 5 to the 3rd, 7 to the 5th does in fact divide 140 raised to the 10th power. How about a different problem? 112 divides 36 to the 15th, yes or no. And again, I want to write all of my numbers as products of primes. 112, 2 to the 4th times 7, 36 to the 15th works out to be 2 to the 30th times 3 to the 30th. And remember that one number divides another only when I can write the second number as a product of the potential divisor and something else. So I'd like to write this, 36 to the 15th, as a product of 2 to the 4th times 7, and whatever's left over. However, there's a problem. 112 requires a 7. I didn't have a 7 in here, so there's no way that I can write this as a product of this and something else. So because of the prime factorizations, because I need a 7 to make 112 and I don't have it, then 112 does not in fact divide 36 raised to the power of 15.