 Once you've studied prime numbers, this leads us to a very important result known as the fundamental theorem of arithmetic. And this goes back to the idea of prime factorization. So let's try and express 360, not just to find a prime factor, but let's find 360 as a product of prime numbers. So we might begin by writing 360 as a product of any two factors, for example, 36 and 10. And again, we go through the same process we did before, is 10 prime, or can I write it as a product of two other numbers? In this case, I find that 10 is a product of two times five. Likewise, I can ask the question, is 36 prime, or can I write it as a product of two other numbers? And so here I might say, ah, I know 36 is 6 times 6, so I'll split 36 this way. And again, I look at my numbers, 2 and 5 are both prime, so they can't be broken down any further. But on the other hand, 6 can be written as a product of 2 and 3, and likewise, this other 6 as well, also a product of 2 and 3. And again, I look at my numbers, 2 is prime, 3 is prime, and 5 is prime. Now, this structure that we have here is sometimes called a factor tree, and it's a convenient way of representing a sequence of equations. So in particular, it's good to represent the following sequence of equations. Our first step here, 360, I've split into 36 and 10, that's 360 is 36 times 10. I split the 10 into 2 times 5, so that 10 becomes a 2 times 5, the 36 becomes a 6 times 6, and each of these 6's becomes a 2 times 3, and a 2 times 3. So my 360, I've now expressed as a product of prime numbers. And I might do two things to make this a little bit more organized. First off, I'm going to use exponential notation, so I have 1, 2, 3 factors of 2, 1, 2, 2 factors of 3, and a 5. And so I'll write my prime factorization, 360 is 3 2's multiplied together, 2 3's times 5. So here's a question to think about. I got this factorization using this particular factor tree. What if I used a different one? So suppose I do my factorization in a different way. And for example, I might look at 360 and consider that it's, well, how about 8 times 45? And nothing in the rule says my branch is only half to half 2 factors. I might look at 8 and say, ah, I know what that is, that's 2 times 2 times 2. And then maybe I'll split this 45 and think a moment, what, multiplies to 45? Well, that's 5 times 9. And again, my 2's and 5's are prime, the 9 is not, so I'll split the 9 into 3 and 3. And so I have a totally different factor tree. I can still interpret it in the same way. 360 is 8 times 45. And then each of these 8's is 2 times 2 times 2. 45 is 5 times 9. 9 is 3 times 3. And so there's my product, 360. And I can again express this in exponential notation. It's 3 2's, 2 3's, and a 5, all multiplied together. And what you might notice is that this product here is the same as the product we found using the other factor tree. And the fact that the two prime factorizations are identical is actually an example of what's known as the fundamental theorem of arithmetic, which is any composite number can be expressed uniquely as a product of primes. In some sense, the primes that you use to multiply to the composite number are the recipe for constructing that number. And if you change the number of primes, if you use different primes, you're using a different recipe and you're getting a different number. So for example, let's consider this problem. Let's consider two numbers. n is 2 to the 16th, and m is 2 times 3 to the 8th times 5. And we want to prove or disprove that the two numbers are equal. Well, the hard way, this is just 16 2's multiplied together, so I can figure out what that is. This number here is 2 multiplied by 8 3's, then multiplied by 5, and I can figure out what that number is. And I can stare at these two numbers and say, ah, they are different numbers. Now obtaining these two numbers, having to multiply those 16 2's and the 2 and the 8 3's and the 5, that's going to take a bit of time. If I were doing this by hand, that would probably take me about half an hour to get at the wrong answer. But we could do it that way if we really wanted to, and if we had lots and lots of time on our hands and we really didn't have anything better to do. On the other hand, there's an easier way of looking at that. The fundamental theorem of arithmetic says if the recipe is different, the numbers are different. And so I look at the recipe. n has 16 2's, m does not. So m and n have to be different numbers. Now the important requirement for the fundamental theorem of arithmetic is that the factors have to be prime numbers. So let's consider this problem. Let n be 27 to the 15th times 25 to the 18th and m be this thing. And let's see if n is equal to m. So again, let's do this the hard way. So n is 15 27's multiplied together and then multiplied by 18 25's. So I can do that in about a week from now. I'll come back and tell you that the answer is this. So here's what I get when I multiply all those numbers together, this huge number here. And well, I haven't done this yet, so I'll multiply together 30 15's and 6 45's times 27. And I'll do that in about a week from now. I'll come back and tell you that this is the answer. And then I'll stare at these two numbers and my eyes are probably going to blur over at some point. But it looks like these two numbers are actually the same. So about two weeks after you ask this question, we come back and we say, yeah, these two numbers are in fact the same number. Well, let's see if we can do this a little bit more easily. So if I want to use the fundamental theorem, the important thing here is I need to have the numbers expressed as products of primes. So here this number 27 is 3 to the third, this number 25, 5 to the second, and I could use my rules of exponents. There's three 3's in here and there's 15 sets, so there's 45 3's altogether. There's two 5's in this set and there's 18 of these sets, so there's 36 5's altogether. So this number N, 45 3's, 36 5's all multiplied together. Likewise, if I consider this number M, I can rewrite this number as a product of prime factors. 15 is 5 times 3. 45 is 3 to the second times 5. 27 again, 3 to the third. So here's my number 15 to the 30th, 45 to the sixth, 27. And I have, let's see, that's 33's, 12 more 3's, and 3 more. So that's 30, 12, and 3. That's 45 3's, 35's, and 6 5's, 36 5's. So this is going to be 45 3's, 36 5's, and now I can check the two numbers. And I have the same number of 3's, the same number of 5's, and the same prime factors and the same numbers. The recipes for the two numbers are the same, so the two numbers themselves are the same. Well, okay, you do have to do the factorization. You do have to do the expansion of the exponent. So if you really want to, you can compare the two numbers after you multiply them out. But it's probably easier just to do the prime factorization of the two numbers. Either way works.