Least common multiple
What is the least common multiple of 36 and 12? So another way to say this is LCM, in parentheses, 36 to 12. And this is literally saying what's the least common multiple of 36 and 12? Well, this one might pop out at you, because 36 itself is a multiple of 12. And 36 is also a multiple of 36. It's 1 times 36. So the smallest number that is both a multiple of 36 and 12-- because 36 is a multiple of 12-- is actually 36. There we go. Let's do a couple more of these. That one was too easy. What is the least common multiple of 18 and 12? And they just state this with a different notation. The least common multiple of 18 and 12 is equal to question mark. So let's think about this a little bit. So there's a couple of ways you can think about-- so let's just write down our numbers that we care about. We care about 18, and we care about 12. So there's two ways that we could approach this. One is the prime factorization approach. We can take the prime factorization of both of these numbers and then construct the smallest number whose prime factorization has all of the ingredients of both of these numbers, and that will be the least common multiple. So let's do that. 18 is 2 times 9, which is the same thing as 2 times 3 times 3, or 18 is 2 times 9. 9 is 3 times 3. So we could write 18 is equal to 2 times 3 times 3. That's its prime factorization. 12 is 2 times 6. 6 is 2 times 3. So 12 is equal to 2 times 2 times 3. Now, the least common multiple of 18 and 12-- let me write this down-- so the least common multiple of 18 and 12 is going to have to have enough prime factors to cover both of these numbers and no more, because we want the least common multiple or the smallest common multiple. So let's think about it. Well, it needs to have at least 1, 2, a 3 and a 3 in order to be divisible by 18. So let's write that down. So we have to have a 2 times 3 times 3. This makes it divisible by 18. If you multiply this out, you actually get 18. And now let's look at the 12. So this part right over here-- let me make it clear. This part right over here is the part that makes up 18, makes it divisible by 18. And then let's see. 12, we need two 2's and a 3. Well, we already have one 3, so our 3 is taken care of. We have one 2, so this 2 is taken care of. But we don't have two 2s's. So we need another 2 here. So, notice, now this number right over here has a 2 times 2 times 3 in it, or it has a 12 in it, and it has a 2 times 3 times 3, or an 18 in it. So this right over here is the least common multiple of 18 and 12. If we multiply it out, so 2 times 2 is 4. 4 times 3 is 12. 12 times 3 is equal to 36. And we are done. Now, the other way you could've done it is what I would say just the brute force method of just looking at the multiples of these numbers. You would say, well, let's see. The multiples of 18 are 18, 36, and I could keep going higher and higher, 54. And I could keep going. And the multiples of 12 are 12, 24, 36. And immediately I say, well, I don't have to go any further. I already found a multiple of both, and this is the smallest multiple of both. It is 36. You might say, hey, why would I ever do this one right over here as opposed to this one? A couple of reasons. This one, you're kind of-- it's fun, because you're actually decomposing the number and then building it back up. And also, this is a better way, especially if you're doing it with really, really large and hairy numbers. Really, really, really large and hairy numbers where you keep trying to find all the multiples, you might have to go pretty far to actually figure out what their least common multiple is. Here, you'll be able to do it a little bit more systematically, and you'll know what you're doing.