 One of the key concepts in number theory is the idea of a prime and a composite number. And so these come about as follows. So let n be some integer that's greater than one. One is a special case for reasons that we'll discuss later. And if I can find two positive integers, p and q, both of which are less than n, where the product is equal to n, we'll say that n is a composite number, and that p and q are factors of n. On the other hand, if we can't find such integers, we'll say that n is a prime number. Now, for example, this can actually be a little bit more challenging than it might seem to be. So, for example, let's try to identify as prime or composite the number 4. And what's going to make this somewhat more challenging is we'll actually want to try and prove our answer. Now, if it turns out that 4 is composite, since I could write 4 as 2 times 2, I can write 4 as the product of two smaller integers, which is all I need to satisfy the definition of what it means for a number to be composite. When we actually run into difficulties is when we try to prove that a number is prime. So, for example, let's consider the number 5. We want to identify it as prime or composite. We want to prove our answer. Now, remember that in order for a number to be identified as prime, we need to show that it cannot be written as the product of smaller integers. Well, that's kind of a problem because how do we know it can't be written as a product of smaller integers? Well, we have to kind of go through our list of integers and see what it could be. So, let's try it out. So, we'll form our list of products of smaller integers, and it turns out that there's six different possibilities, and we'll check them all out, and we'll see that none of these products is equal to 5, and so that does allow us to conclude that 5 cannot be written as the product of smaller integers. 5 must be a prime number. Now, you might say, well, wait a minute, that's not how we prove numbers were prime when we were in elementary school, and the answer is actually it was. You didn't quite do it in this fashion, but the process that you did use to find whether a number was prime or not is basically an analysis of any product of smaller numbers and the determination that no product of smaller numbers will actually make 5. We'll discuss this a little bit more later, but this has an important tie-in to what's called the fundamental theorem of arithmetic, which is to say any integer greater than 1 is either prime or can be expressed as a unique product of primes, and what that uniqueness means is that I can change the order around and I don't get a different product. What I get is that same product just in a slightly different order. Now, the reason that we want to exclude 1 from consideration is that because multiplication by 1 doesn't change the value, then this product here, 2 times 3 times 5, is the same as 3 times 5 times 2 times 1 times 1 times 1, and if I allowed 1 to either be prime or composite, I would have to take into consideration the fact that I could add on an extended string of 1s to any product that I wanted to without changing the product, and that would make the fundamental theorem of arithmetic a little bit clunkier because then we'd have to say, well, except for factors of 1. We have to do that later on when we look at more general numbers systems, but for the integers, we will ignore 1 as a potential factor for now. And so, for example, here's how we might use the fundamental theorem. So I have two numbers. n is 3 to the power of 5 times 7 to the power of 20, and m is 35 to the power of 50, and I want to prove or disprove that n and m are equal to each other. What does this require? Well, by the fundamental theorem of arithmetic, both n and m can be expressed as a unique product of primes, and we won't go into the details here, but 3 and 7 are both prime numbers, so n is already expressed as a product of prime numbers. 5 3s, 27s, multiply them all together, you get n. m, on the other hand, 35 is not a prime number, so I can express m as a product of primes. I just haven't since I'm using 35. So first thing I want to do is I want to write 35 as a product of prime numbers. That's 5 times 7, and standard rules of exponents tell me that m is 5 to the 50th, 7 to the 50th, and now I'll compare my two numbers. The fundamental theorem of arithmetic says that if I have a different product of primes, I must have a different number, and here I can see that my products of primes are different. I have 5 3s versus 55s, 27s, 57s. I have different primes, I have different numbers of primes, so the two numbers must be different numbers.