 So another topic related to factorization is that of the least common multiple of whole numbers. So there's a bit of unfairness here. In both the definition for divisor and the definition for factor, we talk about a and b, but we haven't said anything about n. So let's remedy that. It's also useful to define a multiple. Suppose n is equal to a times b. Then we say that n is a multiple of a or a multiple of b. So let's find a multiple of 72. And again, definitions are the whole of mathematics. All else is commentary. If I want to find a multiple, let's pull in our definition of a multiple. So if I want to find a multiple of a number, I need to write that number times something else. So we find n is 72 times something. And our definition doesn't specify what our something has to be. So for example, if b is equal to 2, equals means replaceable. So I'll replace and multiply. And so 144 is a multiple of 72. Once we have the notion of a multiple of a number, we can also talk about the idea of a common multiple of several numbers. Given two or more numbers, a common multiple is a number that is a multiple of all the given numbers. For example, suppose we want to find a common multiple of 30 and 75. Definitions are the whole of mathematics. All else is commentary. So first of all, we want a number that's a multiple of 30. So n has to be 30 times something. We also want the number to be a multiple of 75. So we know that n has to be 75 times something. Equals means replaceable. So anytime we see n, we can replace it with 30a. And so this means 30a must be 75b. And now for the tricky part of this, if we let a be 75 and b be 30, we get 30 times 75 equals 75 times 30, which is true. And so that says that n equals 30 times 75 is a common multiple of both 30 and 75. We can multiply that out and get n equals 2250, and this suggests the following. Given any numbers a and b, a common multiple is n equal to the product of a and b. Now, while we can always find a common multiple by multiplying our numbers together, this gives us large numbers. Now remember, there's nothing wrong with large numbers, and we should be comfortable working with them. But sometimes we like to deal with smaller numbers. And this leads to the following idea. We define the least common multiple as follows. Given two or more numbers, the least common multiple is the smallest number that is a multiple of all the given numbers. So let's try to find the LCM of 30 and 75. So again, remember, if we're just looking for a common multiple, we can always find a common multiple by multiplying all of our numbers together. But because this problem specifically asks us to find the least common multiple, we'll have to find the least number. So one way we can do it is this. First, we'll list all multiples of 30 and 75. So we want to find every number that's 30 times something and every number that's 75 times something. Now, that is a bit of a problem since there's no end to the multiples. So rather than listing every possible multiple of 30 and then every possible multiple of 75, let's step through them one by one. So a multiple of 30 is 60, then 90. And since 90 is more than 75, let's take a look at our next multiple of 75, which is 150. And since 150 is more than 90, we'll take a look at our next couple of multiples of 30, which are, and we see that 150 is a multiple of both numbers. And because we've listed all of the smaller multiples, we also know that 150 is the smallest number that is a multiple of both. Or we can do this a second way. Remember, the fundamental theorem of arithmetic tells us that the prime factors of a number give us a recipe for the number. So let's find the recipe for 30 and for 75 and see what we can do with that. So let's factor 30 into primes. We can also factor 75 into primes. And let's think about this. Since 30 is 2 times 3 times 5, any multiple of 30 must include a 2, a 3, and a 5. And likewise, since 75 is 3 times 5 times 5, any multiple of 75 must include a 3 and 2 5s. And what that means is if you're a multiple of both, you have to include a 2, a 3, and a 5. Well, actually just having 1 5 will not guarantee you're a multiple of 75. You actually need 2 5s. And if you want to be the LCM, you contain nothing else. And so the LCM is going to be 2 times 3 times 5 times 5. And that guarantees that you're a multiple of 30 because you include a 2, a 3, and a 5. And you're also a multiple of 75 because you include a 3 and 2 5s. And at this point, it's worth introducing an idea. We already have our LCM in factored form. And while we could multiply it out, factored form is the best form to leave things in. And part of that is because factoring is so hard, if we have a factorization, we don't want to lose track of it. We can organize this in another way. We've already factored 30 and 75. Let's go ahead and list those factors in a somewhat organized fashion. So 30 is 2 times 3 times 5. And 75 is 3 times 5 times 5 where we'll line up the existing factors. And now to find the LCM, we'll just carry all of our factors straight down. We need the 2, we need the 3, we need the 5, and we need the other 5. And there's our LCM. So now we have two ways of finding the LCM. We can find all the multiples or we can factor. And you might ask yourself, why bother to learn more than one way to find the LCM? And the answer to that is that if you only have a hammer, every problem must be treated like a nail. And that's OK if you're trying to pound a nail. Less useful if you have some screws. And you might not want to try to open a jar of pickles this way. So for example, if I want to find the LCM of 49 and 98, I might list multiples of 49 and 98. So 49 has a multiple of 98 and whoa, that's the LCM. Or we can try to factor 49 factors as 98 factors as and carrying down the factors gives us the LCM.