 So another useful comparative property of numbers is known as the LCM, the least common multiple, and this emerges from the following. If I have two or more numbers, the least common multiple is the least, the smallest number that's a multiple of all of them. Mathematicians are not particularly noted for their ability to create novel and exciting names for things. So the least common multiple is the least multiple. Because we know the definitions, it's actually easy to do the mathematics, given any number p, a multiple can be found by multiplying p by any whole number q. So given any number, the product p times q is a multiple of p, and again, if I have two or more numbers, I want to find the number that is the smallest multiple of all of them. So for example, let's find the least common multiple of p equals something, and q equals something else, conveniently in our prime-factor form. So if I want to find a common multiple of p and q, I want to find a number that is a multiple of both, and the simplest possible multiple of both p and q is, well, I want something that's a multiple of p, so it's something times p, and I want something that's a multiple of q, so it should be q times something. Well, how about p times q? This is a multiple of p because it's p times something, and it's a multiple of q because it's something times q. So a common multiple of p and q is the product of p and q. Now, this might not be the least common multiple, so let's eliminate what we don't need. Now, how you speak influences how you think. So in order to be a multiple of p, what I need is at least three factors of two, because that two to the third is part of my factorization. So I need at least three factors of two, and in order to be a multiple of q, I need at least, well, I don't need any multiples of two because two does not appear in the prime factorization. So I need at least three factors of two to be in the least common multiple. So I've got to keep this two to the power of three. Now, same argument in order to be a multiple of p, I need at least two factors of three, and to be a multiple of q, I need at least three factors of three. So in order to be a multiple of both of them, I need at least two, at least three, I need at least three factors of three. So I need this three to the third, and I don't really need this three to the second, that is, that is a bit, that's actually unnecessary, but I need at least a three to the third. And finally, in order to be a multiple of both p and q, well, in order to be a multiple of q, I need at least two factors of five, and I don't need any factors of five to be a multiple of p, but I can include them anyway, so I need these five to the second. And so in order to be a multiple of all of these things, what I need is at least three factors of two, at least three factors of three, and at least two factors of five. And we've eliminated all the factors we don't actually need, and so that says what we have left is going to be our least common multiple. Well, here's another common problem. So hot dogs come in packages of ten, while hot dog buns come in packages of eight. You're mileage may vary on these, but let's assume that these numbers are correct. What's the smallest number of packages of each that you can buy, so you will have the same number of hot dog buns as hot dogs? Now, a little analysis goes a long way, so let's think about this. If I buy more than one package of hot dogs, then the number of hot dogs I have is going to be a multiple of ten. Likewise, if I buy more than one package of hot dog buns, the number of hot dog buns is going to be a multiple of eight. And since I want to have the same number of hot dog buns as hot dogs, then I want to find a multiple of ten that is also a multiple of eight. And because I want to find the smallest number of packages, I want to find the smallest number of, let's say, a multiple of eight and a multiple of ten. I want to find the least common multiple of ten and eight. Let's go ahead and do that. To find the LCM of ten and eight, we should factor both. Ten equals two times five. Eight equals two to the third. And I can always find a common multiple by multiplying the two numbers. So a common multiple of ten and eight is ten times eight. So there's my ten times eight. Well, that's not necessarily the least common multiple, so we'll eliminate the things that we don't need. So in order to be a multiple of ten, I need to include a two and a five. In order to be a multiple of ten, I need to include a two and to be a multiple of eight I need to include three factors of two. So I need at least three factors of two. So, to be a multiple of both, I need at least three factors of 2, and this one, this extra one here, I don't really need, so I can get rid of it. If I want to be a multiple of both, I need at least one factor of 5. I don't need any here, but I need at least a factor of 5 here, and so I'm going to keep this factor of 5 here, and so there's my common multiple of both. And so there's my least common multiple. Now be careful with what we've found. This is the least common multiple of 10, the number of hot dog bucks, and 8, the number of hot dog buns. However, the question asks for the smallest number of packages. So let's think about that. So our least common multiple of 5 times 2 to the third, well that is a multiple of 10 as promised. It's also a multiple of 8 as promised. And so this says that I need 2 to the second, I need 4 packages of 10, that's 4 packages of hot dogs, and I need 5 packages of 8, that's hot dog buns. So I need 4 packages of hot dogs, and 5 packages of hot dog buns.