Least Common Multiple (LCM)
The Least Common Multiple, or LCM, of two or more numbers is the smallest number other than zero that's a multiple of each number. In other words, the LCM is the smallest number they all divide into evenly. There are two ways to find the LCM.
Method 1 is to make a list of multiples for each number till we find a multiple on both lists. To find the LCM of 4 and 6, we list a few multiples of 4 and a few multiples of 6 and then check to see if they have a multiple in common. If not, we continue on with our list until we find one that is. We start our list with the number itself.
Multiples of 4: 4, 8, 12, 16, 20, 24…
Multiples of 6: 6, 12, 18, 24…
Aha! There's more than one multiple that shows up in both lists. It's a conspiracy, and it looks like either 12 or 24 could be the culprit. Since 12 is less than 24, it's definitely the least common multiple.
Method 2 is great when we're finding the LCM of larger numbers such as 28 and 40. To find the LCM, we list the prime factors of both numbers.
28 = 2 × 2 × 7
40 = 2 × 2 × 2 × 5
Now we grab every single factor in either of the lists. So we definitely grab 2, and also 5, and 7. That's everything, but how many do we grab? Whichever number has the most of a given factor: that's how many. Since 28 only has two 2s, and 40 has three 2s, we grab three 2s. There's only one 5 so we just grab a 5, and there's only one 7 so we grab a 7. Now multiply them all together.
2 × 2 × 2 × 5 × 7 = 280
280 is the smallest number that both 28 and 40 divide into evenly.
Recap: to find the LCM of two numbers, multiply all the prime factors using the greatest number of times the prime factor appears in either number.
Greatest Common Factor (GCF)
The Greatest Common Factor, or GCF, of two or more numbers is the largest number that's a factor of each number. In other words, it's the biggest number that divides into all of them evenly. There are two methods to find the GCF as well.
Method 1 is a good method for smaller numbers because it's simpler to remember, but it can lead to more mistakes with larger numbers because it's easy to forget factors.
To find the GCF, list all the factors of each number—not just the prime factors, but any number that divides into it evenly. It helps to write these factors in pairs. Then compare the factors of each number until you find the largest one that they both share.
To find the GCF of 28 and 32, we list pairs of factors. Wait, factors come in pairs? Yup, any two numbers that multiply to 28 are a pair of factors. And if we write them on opposite ends of a line, working toward the middle, we can make sure we get them all.
28: 1, , 28
32: 1, , 32
Here's the first pair of factors. Next up: 2. We know 2 × 14 = 28, and we know 2 × 16 = 32.
28: 1, 2, , 14, 28
32: 1, 2, , 16, 32
Since 3 doesn't divide evenly into either number, we move on to 4.
28: 1, 2, 4, 7, 14, 28
32: 1, 2, 4, 8, 16, 32
Next in the lists come 5 and 6, but neither divide evenly into 28 or 32 and we already have 7 and 8 on our list, so we know we're done.
Well, whaddaya know, the greatest factor that appears in both lists is 4, so the GCF of 28 and 32 is 4. Badda bing.
In Method 2, we start by listing the prime factors for each number. Then we multiply all prime factors that show up in both lists. But how many of each? The GCF wants to be different from its cousin the LCM, so for the GCF, we use as many prime factors as are in both lists.
To find the GCF of 90 and 252, we list the prime factors:
90 = 2 × 3 × 3 × 5
252 = 2 × 2 × 3 × 3 × 7
Now we find which factors appear in both lists. And the winners are: 2, 3, 3. We don't include 5 because it's not in both lists, and we don't include 7 for the same reason. Notice we do include two 3s because both lists have at least two 3s.
So the GCF is 2 × 3 × 3 = 18. In other words, 18 is the biggest number that will divide evenly into both 90 and 252.