Let’s look at least common multiple, what it is, and how we would find it. Least common multiple, well first it’s a multiple. A multiple is any number multiplied by an integer. Or put another way, a number that may be divided by another number with no remainder.
Here are some examples of some multiples. Multiples of three are three, six, nine, twelve, fifteen. Multiples of twelve would be twelve, twenty-four, thirty-six, forty-eight.
But we’re not just looking for the multiple, we’re looking for something that is common. It’s shared by two or more numbers. It’s also going to be the least, and by that we mean the lowest number.
So we would say the least common multiple is the smallest number that is a multiple of two or more numbers. Here’s an example: find the least common multiple of four and six. When we’re working with least common multiple, we usually write LCM, open parentheses, then we enter the numbers that we wanna find the least common multiple of. That’s the format that you’ll usually find the least common multiple in.
To solve this problem, we wanna find the prime factorization of both four and six. Four is made up of two times two, and six is made up of two times three.
For something to be a multiple of four, it needs two and a two. So whatever the least common multiple is, it’s going to need two times two in it.
For something to be a multiple of six it needs a two. And we have already included one two, so we don’t have to add another one. But we haven’t added a three yet, so we’ll need to do that. Two times two times three will be our least common multiple of four and six. We know that this number will be a multiple of four because it has two times two in the factorization, and we know that it’ll be a multiple of six because it has three times two in the factorization.
So our least common multiple here is twelve. That means that twelve is the smallest common multiple of four and six. So maybe you’re thinking, “Well I could have figured that out with all of that factoring.” That’s true. There is another way to do it.
You could list all the multiples of both of those and then try to find the smallest one. And here, that’s twelve. But it’s not always the best way. The reason that’s not always best is you could end up with a problem like this, the least common multiple of one hundred and thirty-seven, fifty-three, and five hundred and seventeen. You certainly wouldn’t wanna waste your time listing out each of the multiples. Instead, it’s always best to look for the prime factors and then find the multiple.
Speaking of bigger numbers, here’s a few that are slightly bigger. What’s the least common multiple of thirty-nine and thirty-two? We wanna solve this problem by factoring so let’s start there. I recognize that thirty-nine is divisible by three; thirty-nine is three times thirteen. Both of those are prime numbers so we’ll stop there. Thirty-two is divisible by two and sixteen, sixteen is two times eight, eight is two times four, and four is two times two.
When you look at the factorization of thirty-nine and thirty-two, you’re going to realize that they actually share no common factors, which means to find a multiple that they have in common, we need to multiply three times thirteen times two times two times two times two times two. Or you can also just multiply thirty-nine times thirty-two. The least common multiple for thirty-nine and thirty-two is one thousand two hundred and forty-eight.
We’re going to use the same process to find the least common multiple of three numbers that we did for two. We need to find the factorization of all three of these numbers. Let’s start with the four. That’s two times two. And then the six, two times three. And finally with forty-nine, we find it’s seven times seven.
We know we’ll need two times two. Since we’ve already added a two, we now only need to add a three for our six. But the factors of forty-nine are two sevens, and we’ll need to include both of those.
The least common multiple of four, six, and forty-nine will be two times two times three times seven times seven for a final answer of five hundred and eighty-eight.
Here’s our last example, the FIFA World Cup takes place every four years and the US census is taken every ten years both of them took place in two thousand and ten. When is the next year in which both of these events will happen again? Let’s pull out the key information here, which is the World Cup takes place every four years and the US census is taken every ten years. So we wanna know what are multiples, what is in fact is the least common multiple of four and ten?
We’ll follow the same procedure, finding the factors of four and ten. We know four is two times two and ten is two times five.
We’ll need two times two and also that five factor. Two times two times five is twenty. The least common multiple of four and ten is twenty, but that’s not the answer to the question.
The question is when is the next year in which both of these events will happen. So we need to know what should we do with that twenty. That twenty means that every twenty years, the four and the ten will line up. So we need to add twenty years to two thousand and ten, which is two thousand and thirty. Two thousand and thirty is the next year that the World Cup and the US census will happen at the same time. So now you know, and you’re ready to use these skills to find the least common multiple of lots of numbers.