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Math 102: College Mathematics14 chapters | 108 lessons

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Lesson Transcript

Instructor:
*Luke Winspur*

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

The least common multiple of two numbers is the smallest number that can be divided evenly by your two original numbers. See some examples of what I'm talking about here!

The least common multiple is a math topic you usually use when you're trying to find a common denominator between two fractions, and it's one of those things that you learn pretty early on in your education, but it can easily be forgotten or mistaken for a different math idea, usually the greatest common factor. So, let's start by reminding you exactly what it is.

The **least common multiple** of two numbers, often written as the **LCM**, is the smallest number that can be divided evenly by those original two numbers. For example, the LCM of 5 and 6 is 30, because it is the smallest number that both 5 and 6 go into. And, that's it - least common multiple.

But, in order to not forget what a least common multiple is in the future, it's probably best to understand some of the vocabulary in the name - mainly, what is a multiple? Well, the multiples of a number are just what you get when you multiply that number by 1, then 2, then 3 and so on. For example, the multiples of 2 are 2, 4, 6, 8, 10, 12 and on and on. Or, the multiples of 7 are 7, 14, 21, 28, 35 and on and on and on. Notice that 2 is a multiple of 2, and 7 is a multiple of 7. That means that any number is a multiple of itself; we'll need to remember that a little bit later on. Anyways, now that we know this vocabulary, we can say that the least common multiple just means the smallest multiple that two numbers have in common.

The most foolproof way to find a least common multiple is to list out all the multiples of each number and then find the first one they have in common, like I've been showing you so far. LCM of 8 and 6? Well, multiples of 8 are 8, 16, 24, 32... and the multiples of 6: 6, 12, 18, 24 - hey! Got it - 24.

But, this way can get kind of annoying, especially when the numbers get bigger, so there are some shortcuts as well. Besides just knowing your times tables really well and being able to do it all in your head, it's often the case that the **least common multiple** of two numbers is just what you get when you multiply those two numbers together.

If we look at the first example we did, the LCM of 5 and 6, the answer was 30, which is exactly what 5 times 6 is. But, the second example we did didn't follow this pattern. The least common multiple of 6 and 8 was 24, which is not equal to 6 times 8. Now, the reason this trick works for the first one but not the second one is the fact that the numbers in the first one do not share any factor.

Factors are kinda like the opposites of multiples. If multiples are the bigger numbers that 6 go into (6, 12, 18, 24...), factors are the smaller numbers that go into 6 (1, 2, 3, 6). For example, the factors of 15 are 1, 3, 5 and 15, because those are all the numbers that you can divide 15 by and get a nice number out. Or, we can say the factors of 22 are 1, 2, 11 and 22.

So, back to the trick. I could just write out all the multiples of 11 and 12 and find the first one they have in common, or I could use the shortcut and just do 11 times 12 to get my answer of 132, because I know that 11 and 12 don't have any factors in common except 1.

But then, when I do another problem, maybe like the least common multiple of 6 and 20, I can't use the shortcut because 6 and 20 share a factor that isn't 1. In this case, it's 2. That means 6 times 20 is not my answer, and I have to instead just list out all the numbers and find the first one they have in common, which appears to be 60.

If you want to get really fancy, you actually could still use the shortcut, but it's a slightly longer shortcut, and after you do 6 times 20, you divide that by whatever factor they have in common, which in this case was 2, so you could end up with the correct answer.

Before we finish, I should tell you that sometimes the LCM of two numbers can be one of the original numbers itself. Remember how we said earlier that a number is a multiple of itself? Well, that means that, for example, if we were doing the LCM of 3 and 9, we'd just get 9. Or, if we were doing the LCM of 4 and 16, it's just 16. So, if you see this happen in one of your problems, don't worry, totally okay.

Let's quickly review what we've learned. The multiples of a number are the numbers you get when you times that first number by 1, then by 2, and then 3 and 4 and so on. The **least common multiple** of any two numbers is the smallest multiple that they have in common, and it's also written as the **LCM** for short. To find the LCM of any two numbers, you can write out a list of their multiples until you find one that's the same. Or, there is a shortcut that says as long as they don't have any factors in common, you can just multiply those two numbers together.

Finally, as a side note, the place where you'll probably find yourself using the LCM most often is when you are trying to find a common denominator between two fractions.

Following this lesson, you'll be able to:

- Define least common multiple and explain how to find it
- Differentiate between factor and multiple
- Identify a shortcut to finding the least common multiple in a particular situation
- Explain when you will likely need to find the least common multiple

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