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.
LCM of Multiples
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.
Lesson Summary
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.
Learning Outcomes
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