# Multiples

A multiple is a product of a number and an integer, or more simply, when you multiply two whole numbers together. Every number has an infinite number of multiples. For example, the multiples of 2 are 2, 4, 6, 8, 10, 12, 14 and so on, to infinity. You may be asked to make a list of multiples until you hit a certain number (for example, list the multiples of 2 from 2 to 50) or you might be asked to list a certain number of multiples (for example, list the first 5 multiples of 4). The important part here is that you carefully read the directions and make sure that you understand what the question is asking of you before you answer. Here are several examples:

List the first five multiples of the following numbers: 3, 6, 9, 10, 7, 12, 50

3: 3, 6, 9, 12, 15

6: 6, 12, 18, 24, 30

9: 9, 18, 27, 36, 45

10: 10, 20, 30, 40, 50

7: 7, 14, 21, 28, 35

12: 12, 24, 36, 48, 60

50: 50, 100, 150, 200, 250

You also might recognize multiples as "counting by" the above listed numbers.

## Least Common Multiple

Once you learn what multiples are and how to find them, you will often be asked to find the "least common multiple" between two numbers. For example, this is how you find the least common multiple (LCM) between 4 and 5:

First, list the multiples of each number. Now, this could go on until infinity, so we recommend only listing the first 5 multiples of each number, and then continuing on if you need to.

4: 4, 8, 12, 16, 20

5: 5, 10, 15, 20, 25

Next, underline (or circle) any factors they have in common. If they don’t have common factors, you need to keep going (check the next five multiples). If they do have common multiples, underline them, like this:

4: 4, 8, 12, 16, __20__

5: 5, 10, 15, __20__, 25

Now, you have to find the least common multiple. In our example, we only have one common multiple, so that is our LCM. However, if we had more than one common multiple, we would pick the smallest one. Therefore, our LCM between 4 and 5 is 20.

## When Would You Use LCM's?

Very rarely will you be asked to simply find the LCM between two numbers. However, this process is particularly useful in finding common denominators. For example, let’s say we had the problem 5/8 + 2/5. We would first need to find a common denominator. Here’s how we would use LCMs to find the common denominator.

First, we need to list the multiples of each of the denominators in the problem, 8 and 5.

8: 8, 16, 24, 32, 40

5: 5, 10, 15, 20, 25

At this point, we don’t see any common multiples, so we’ll list 5 more for each one:

8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

Now, we underline the common multiples, like this:

8: 8, 16, 24, 32, __40__, 48, 56, 64, 72, 80

5: 5, 10, 15, 20, 25, 30, 35, __40__, 45, 50

We notice that 40 is our only common multiple, thus for this problem it is our LCM as well. Next, you would use the LCM as your new common denominator between the two fractions. You would expand each fraction so that it has a denominator of 40, like this:

Then, you would just add the two fractions together: 25/40 + 16/40 = 41/40. Then, you would convert your fraction into a mixed number (since the fraction is improper) and you would end up with 1 1/40.

Let’s try one more example. This time, you can try solving it on your own, and then type your answer into the answer box to see if you’re right!

First, list the multiples of both denominators of the fractions, like this:

9: 9, 18, 27, 36, 45

12: 12, 24, 36, 48, 60

Next, underline any common multiples:

9: 9, 18, 27, __36__, 45

12: 12, 24, __36__, 48, 60

We notice that 36 is our only common multiple so far, so it is also our LCM. Next, you would have to use the LCM as your common denominator, and expand both of the original fractions to include a denominator of 36, like this:

Now, you can take the expanded fractions and subtract them, like this: 28/36 – 15/36 = 13/36. This fraction cannot be reduced, so you’re done and your final answer is 13/36.