In this lesson, we're going to find the least common multiple between two numbers. So what is a multiple? Multiples of a number are the product of the number and a natural number. Remember, the natural numbers are accounting numbers, 1, 2, 3, 4 and so on. So if we want to find the multiples of the number three, then we're going to do three times one, three times two, three times three, and we can go on and on and on. So that's one way of finding the multiples of a number. Another way is to start off with the number and then skip count by that number. So start off with the number three and then just keep adding three to that number and keep going. So we can end up with three, add another three and get six. Add another three and get 9, 12, 15, 18 and we can go on and on for as long as we need to go. Just keep adding three each time. When we want to find the least common multiple, we're looking for the smallest multiple that the numbers have in common. So let's find the least common multiple of twelve and 18. First, we list the multiples of twelve where we can start with the number twelve and then just keep adding twelve. So we get twelve, 24, 36, 48, 60, 72, and we can keep going on and on if we like. And then we list the multiples of 18. Start with 18 and then keep adding 18 to it -  18, 36, 54, 72, 80, 98. Now, our task is to find the least common multiple between those numbers. So let's see what multiples they do have in common. We can see that both numbers have a 36 and a 72, but we want to find the least or the smallest multiple that they have in common, which will be the 36. So we would say that 36 is the least common multiple between twelve and 18. Now of course in math there's normally another way that we can solve the problem. So this time we're going to find the least common multiple, but using prime factorization. So if we list out the prime factorization of twelve, we get two times two, times three. For 18, we get two times three, times three. Now remember, you can always use a factor tree to help you set up the prime factorization of a number. Our next step is to ask ourselves what is the most that each number appears. So for example, we can see that both numbers have two listed in their prime factorization. But we want to see the most that the number two shows up. It shows up twice for the number twelve. So we need to count both of those two. And now we can also see that there's a three in there. Both the numbers have a three in common, but we want to see the most that the number appears in one of the prime factorization. 18 has three showing up twice. So that means that we need to count three two times as well. So we're going to have two twos and two threes. And now we list those out two times two, times three, times three, and then we multiply. We want to find the product of all these numbers, and it would give us 36, which is the least common multiple of twelve and 18. So if we want to find the least common multiple of ten and 45 using prime factorization, we set up a similar process. Ten can be written as two times five, and 45 will be written as three times three times five. Now we ask ourselves what is the most that each number appears in those prime factorizations? The number two shows up just one time, so we need to count that two. The number five shows up once in each number, so we can only count that one once as well. The number three shows up twice, and the number 45. It's okay that three doesn't show up for number ten. Remember, we want to see the most that that number shows up for either list of the prime numbers. So we need to count a two, a five, and both of those threes list those out, multiply them together, and we get 90. So now we know that 90 is the least common multiple of ten and 45. And we would get the same answer by listing out the multiples like we did in our other example as well. So you have two different ways that you can find the least common multiple. So to review, the least common multiple is the smallest multiple that the numbers have in common. Our first method was to list the multiples of each number and then find the smallest multiple that they have in common. Our second method was to list the prime factorization of each number and then look for the most that each factor appears in each list of the prime factorizations, and then multiply all of those factors together. And that will give you the least common multiple.