 So let's talk about one of the key classifications of numbers. And in general, we classify numbers into prime and composite. And so we're going to make the definition. First off, we're going to define the factors of a number. Suppose I have a whole number n that can be expressed as the product of two other whole numbers, p and q. Then I say p and q are called factors of n. And importantly, if p and q are less than n, then they are what we would call the proper factors of n. And this is a very useful definition to keep in mind. Now I'm going to ask a question. We don't actually have to prove the following, but it helps if we think about problems like this one. So here's a number n, and I'm going to write it 33,491 times this other number, 19,871,543. And I want to find two proper factors of n. So let's try to do this as inefficiently as possible. So let's see. Well, I might figure out what this is. And so I'll get out some paper and pencil and do the computation. Actually, I'll go to my favorite search engine here. And so I'll go ahead and multiply 33491 times 1871543. And if I press that, it'll tell me, oh, OK. So here's my product of those two numbers. And so now I want to sit here and stare at this number for a couple of minutes. And try and figure out what can I multiply to get this number? Well, that's a horribly inefficient way of doing this. So maybe we'll be able to do something a little bit better. So let's think about it. Oh, wait a minute. I've already written n as a product of two numbers. You're actually given n as a product of two numbers. And so if you go back to the definition, if I can write n as a product of two numbers, then each of the two numbers are factors. So n is a product of two numbers. So each of these numbers is a factor. And you might make a little bit more thought here. If, by my definition of multiplication, this is this many of these things, then certainly this is smaller than this many of those things. So this number here, 19,871,543, is definitely a proper factor of n. And commutativity allows you to make the same argument. This, 33,491, is also smaller than n. So both of these are, in fact, proper factors. Well, that was an easy warm-up question. So let's take a somewhat more difficult question. So let n be 530 times 36. And I'll find six proper factors of n. Well, here's 2,530, and 36. And again, I can do this using the horribly efficient method. So I'll go ahead and figure out what 530 times 36 is. I get 19,080. And then I sit here and stare for a while at this number and try to figure out what multiplies to that figure. Well, again, somewhat more efficiently. All I really need to do is find two numbers that multiply to n. So if I break these numbers apart, then I can rewrite n as a product of numbers. So for example, 530 times 36. Well, I can keep the 530 and split the 36 into 9 times 4. And so once again, I have n as a product of some things, of 9 and 4 and some other stuff. Or maybe I can try to break the 530 apart into 53 times 10. And so this gives me two more factors, 53 and 10. So now I have my 1, 2, 3, 4, 5, 6 proper factors of my number n. Now this idea of breaking numbers apart by factoring leads to the following important definition, which is of prime and composite numbers. So one important thing, we're going to take whole numbers that are greater than 1. And we're going to call them composite if we can find proper factors other than 1. If we can't find proper factors, then we're going to say that the number is prime. Importantly, we are not defining one to be either prime or composite. And the reason for that we'll see in a moment. So let's consider the problem. We want to prove or disprove that 9 is a prime number. So we either want to show that 9 does have proper factors besides 1 or that no such proper factors exist. And worth noting, we don't know whether 9 is prime or not. We're given a number, and we have to classify it whether it's composite or prime. So we might try to find proper factors, in which case we will not have a prime number. And in this case, 9 is pretty easy to work with. 9 is 3 times 3. And these are proper factors. They're both less than the number. So I have proper factors of 9. And that tells me 9 is not actually a prime number. Now, again, because this is a proved statement, our solution to the problem has to in some way go back to what the definition is. So it might look something like this. Since 9 is 3 times 3, then I have a proper factor. And so 9 is not prime. Actually, it's pretty easy to show that a number is composite. Because all I have to do is to show that I can write it as a product. Primes are more difficult. So here, we're going to prove that 5 is actually a prime number. And so here, we're beginning with the claim that 5 is prime. So let's see if we can prove it. And because it's prime, what I want to do is I want to show that 5 does not have any proper factors other than 1. So let's see. Well, that means that I have to consider every number less than 5 and show that those numbers are not proper factors. So in particular, I have to show that 2, 3, and 4 are not proper factors. Now, again, because the notion of proper factors is part of the definition, I can just make a claim like 2 is not a factor of 5. Hopefully, I'm actually making a true statement here. If this were not true, then my conclusion would be invalid. But if I just want to say 2 is not a factor of 5, as long as I have reason to believe that that statement is true, I don't really have to go into much more detail than this. Again, this is because the concept of factor is part of our definition of the concept of prime. So I can say 2 is not a factor, 3 is not a factor, 4 is not a factor. And so none of the numbers less than 5, with the exception of 1, is a factor of 5. So 5 has no proper factors. And so 5 is a prime number. And again, the important part of the proof that I want to give is the portion that's in green. Now, the proof is actually just a section in green. These are supporting statements. We don't really have to include these as part of the proof. But it's important that they actually be true. So what happens if I consider a number like 127? And if I want to prove that this number is prime? So how would you do that? Well, would you have to do the same thing we did before and verify that no number less than 127 is a proper factor? Well, actually, yes, you have to do this. However, it's possible we might be able to skip some of the lesser numbers. And this leads to a very useful theorem about prime numbers. Any composite number n has a prime factor. So if we want to determine whether a number is composite, we can check to see whether it has prime factors. And the proof of this is actually pretty neat. So we'll go ahead and give a sketch of it. So if n's a composite number, then by definition, I could write n as a product of two numbers, both of which are less than n. And if either of these numbers is prime, we're done. And every now and then, the universe actually gives us an easy problem. And so maybe we do find that one of those numbers is prime and we don't have to go further. But this is often the case. The universe rarely gives us easy problems. So maybe both numbers are composite. Well, what does that mean? Well, if they're composite, that means that I could write both a and b as a product of proper factors. So I could write a, for example, as mn. And because they're proper factors, both of these have to be less than a. And again, we're in a situation, m and n. Well, if either of m or n is prime, we're done. Because then that will be a prime factor of a, which will in turn be a prime factor of n. So if I get two prime numbers or one prime number here, then I'm done with the problem. And again, every now and then, the universe gives us an easy problem. On the other hand, if they're both composite, I can repeat the procedure. And so these are composite numbers, so I can express each of them as a product of two other numbers, both of which are less than m. And so again, if one of these numbers is prime, we're done. If they're both composite, then what I can do is I can continue the problem. And what I end up with is a descending sequence of numbers. I have my initial number n. a is a proper factor, so a is smaller. If a is composite, there's a proper factor m, which has to be smaller than a. If m is composite, there's a proper factor u, which is smaller than m. If u is composite, there's a smaller factor, which is a factor of u. And so on. But because I have a sequence of whole numbers, each of which is smaller than the preceding one, this sequence of whole numbers has to have a last term. Well, the only thing that's going to stop me from finding smaller and smaller numbers is getting one of these numbers eventually to be a prime number. And that's going to be the completion of the fruit. Because sooner or later, I cannot avoid running into a prime number. And once I find that prime number, that's going to be my factor of n. Now, the reason that looking at this proof is useful is that it gives us a somewhat more efficient way of doing a prime factorization. So let's consider the following problem. Find a prime factor of 360. So one thing we might do is we might go through and try and see does two divide, does three divide, and so on. And in some cases, that'll be easy. In other cases, it'll be very tedious and painful. So instead, we can go back to how we did the proof and the proof was find any product of numbers that multiplies to 360. So for example, I might consider 360, 36 times 10. And I check, is this prime, is this prime, or more effectively, can I factor these numbers? And so I might look at 10, for example, and ask myself, well, can I factor 10 or 36? 10's a little easier to factor. So let's go ahead and factor 10. It's 2 times 5. And now I check my numbers. Is 2 prime? And it turns out that 2 is actually a prime number. Likewise, so is 5. And so if I wanted to find a prime factor of 360, I can start with an easy factorization and then continue with the easy factorizations.