 One of the most important results in mathematics is known as the fundamental theorem of arithmetic. And this is based on the following idea. Remember that a whole number n is composite if it could be written as a product of smaller numbers, and a whole number n greater than one is prime if it's not composite. So, if a number isn't prime, then we can write it as a product of smaller numbers, and there are two possibilities. Either every factor is prime or there is a composite factor. But a composite factor can be expressed as a product of smaller numbers. And this leads to the following. Every number greater than one can be expressed as a product of primes. Now for this it helps if you know what the primes are, and unfortunately we're not born with the knowledge of prime numbers. Fortunately they're not too difficult to find, and with a little bit of effort, which I'll put in, we can find the primes under 20, 2, 3, 5, 7, 11, 13, 17, and 19. So let's try to find the prime factorization of 210. So we'll pull in our list of prime numbers under 20, and let's see if that's enough. So if we can write 210 as a product, we can begin our factorization. There's no need to be fancy about the factorization. Anything we can find that multiplies to 210 works, so let's try, oh, how about 10 times 21? And now we want to stare at these numbers 10 and 21, and ask ourselves, self, are these prime numbers? And there's two ways we can do that. First, we can look at our list of primes and see if these numbers are on the list, but that can be a little bit tedious because the list of primes is very, very large. On the other hand, we can ask ourselves, can I write 10 or 21 as a product of smaller numbers? And we have 10 as the product of 2 times 5, and 21 as the product of 3 times 7. Equals means replaceable, so anytime I see 10, I can replace it with 2 times 5, and anytime I see 21, I can replace it with 3 times 7. And again, we can take a look at our numbers and ask ourselves, self, can we write any of these as the product of smaller numbers? Or we could consult our list of primes and see that 2, 5, 3, and 7 are all primes. And so we see that all the factors are primes, so we're done. So another way we can organize these factors is by using what's called a factor tree. And this is a much more graphic way of representing the prime factorization, so let's throw down an MPAA warning. So we might take this number 210 and indicate that it's the product of 10 by 21. Next, we can factor 21 as 3 times 7, and we can factor 10 as 2 times 5. Now the fact that mathematicians call this a tree tells you something about mathematicians, namely, don't have them be your gardeners. But the general organization of this factor tree works something like this. If a number can be written as a product, we write the factors below it, and we keep factoring until we get primes. The roots of the tree are the prime factors. And so we see here, down at the very, very bottom, we have the factors 2, 5, 3, and 7. And this leads to the fundamental theorem. Any whole number greater than 1 is either prime or can be expressed as a product of primes, and, even better, any whole number greater than 1 has a unique expression as a product of primes. And this means that the product of primes is a recipe for the number. So let's take a look, determine whether the statement is true or false. 17 times 9 equal 2 by 2 by 3 by 3 by 3 by 3. Now there's two ways we can do this. First, there's the hard way. We can multiply 17 times 9. We can multiply 2 by 2 by 3 by 3 by 3 by 3. And after all that work, we can see they are not equal. Or there's the easy way. So the fundamental theorem of arithmetic says that any whole number greater than 1 has a unique expression as the product of primes. And for this, it also helps to remember what the primes are. And so again, here's our list of primes under 20. And so the first thing we notice is that all of our factors are prime numbers. And so on the left, we have a recipe for a number. And on the right, we have a recipe for a number. And because the recipes are different, they have to be different numbers. And so we might say the following, since all the factors are prime and the primes on the left are different from the factors on the right, the numbers must be different. Or how about something like this? So we look at the left-hand side and we see that 2, 3, and 5 are all primes. On the right-hand side, we see that 5, 3, and 2 are all primes. And so since both sides are written as products of primes, the fundamental theorem of arithmetic applies. Now if we take a look at the left-hand side, remember the primes are the recipe for the number and this recipe has four 2s, three 3s, and one 5. On the right-hand side, we also have 2s, 3s, and 5s. And in this case, our recipe is three 2s, four 3s, one 5. And while they have the same primes, the number of 2s and 3s is different. And so the numbers are different because remember quantity counts. A large coffee with one sugar is very different from a large coffee with 15 sugars even though they both have coffee and sugar.