# Factors

A factor is a part of a number and two factors multiplied together produce a product. Every number has at least two factors, possibly more. For example, the number 2 has two factors, 1 and 2, because 2 x 1 = 2. Some numbers have more than two factors, like 10. Factors of 10 would include, of course, 1 and 10, but they would also include 2 and 5, because 2 x 5 = 10. Therefore, if we were asked to list factors of 10, we would write: 1, 2, 5, 10.

Let’s practice this. List the factor pairs and factor lists of the following numbers: 15, 20, 17, 43, 28, 32, 99

Solutions—first we’ll list the factor pairs, and then we’ll list the factor list from smallest to largest:

15: 1 x 15, 3 x 5. Answer: 1, 3, 5, 15.

20: 1 x 20, 2 x 10, 4 x 5. Answer: 1, 2, 4, 5, 10, 20.

17: 1 x 17. Answer: 1, 17.

43: 1 x 43. Answer: 1, 43.

28: 1 x 28, 2 x 14, 4 x 7. Answer: 1, 2, 4, 7, 14, 28.

32: 1 x 32, 2 x 16, 4 x 8. Answer: 1, 2, 4, 8, 16, 32.

99: 1 x 99, 3 x 33, 9 x 11. Answer: 1, 3, 9, 11, 33, 99.

Notice that when we wrote our lists, we started by writing the number given to us originally, followed by a colon ( : ). This is standard notation, but not absolutely necessary. Next, we wrote factor pairs. Factor pairs are always the multiplication problems, and look like this: 2 x 1, but do not contain the = or the answer. Last, we listed the factors themselves, separated by commas. You may be asked for just the factor pairs, or just the factor list, or both. Make sure you give the answer that fully completes the question you’re asked.

## Greatest Common Factor

After you learn how to find factors, you may be asked to find the greatest common factor (GCF) between two numbers. The GCF is the largest factor that both numbers share. This means that you would list all the factors for each number. Then, you would circle (or underline, etc) all the factors that the numbers have in common. After that, you would report the greatest number (out of the common—circled—numbers).

For example, find the greatest common factor of 24 and 32. First, list the factors for each number, like this:

24: 1, 2, 3, 4, 6, 8, 12, 24.

32: 1, 2, 4, 8, 16, 32.

Now, go through and circle or underline (we’re going to underline) the common multiples, like this:

24: __1__, __2__, 3, __4__, 6, __8__, 12, 24.

32: __1__, __2__, __4__, __8__, 16, 32.

Now, look only at the underlined numbers. Which number is the biggest? We can see that the greatest number we have on our common factors list is 8, so 8 is our greatest common factor.

Here’s an example for you to try. When you’re done, put your answer in the answer box to find out if you’re right!

The greatest common factor between 75 and 105 is 15. In order to solve this problem, you should have first listed the factors of both 75 and 105, like this:

75: 1, 3, 5, 15, 25, 75.

105: 1, 3, 5, 7, 15, 21, 35, 105.

Then, you would underline the common factors, like this:

75: __1__, __3__, __5__, __15__, 25, 75.

105: __1__, __3__, __5__, 7, __15__, 21, 35, 105.

Last, you would look at all the underlined numbers, and find the greatest number, which in our example is 15. Thus, your GCF is 15.

## When Would You Use GCF's?

Greatest common factors are most often used when reducing fractions. It would be applied like this:

Reduce 15/20.

First, you would list the factors of both 15 and 20, like this:

15: 1, 3, 5, 15.

20: 1, 2, 4, 5, 10, 20.

Now, you would underline the common factors between the two numbers, like this:

15: __1__, 3, __5__, 15.

20: __1__, 2, 4, __5__, 10, 20.

Now, look at the common factors, and find the greatest of the underlined numbers. You will see that 5 is the greatest common factor. This tells you that you can reduce the fraction by 5, like this:

Thus, your reduced fraction is 3/4. Notice that the problem did not directly ask you to find the GCF, but finding it helped you reduce the fraction into lowest terms.

Now, try the next one on your own, and enter your answer into the box to check it when you’re done!

First, you would list the factors of each number, the numerator and the denominator, like this:

18: 1, 2, 3, 6, 9, 18.

24: 1, 2, 3, 4, 6, 8, 12, 24.

Now, underline the factors that these two numbers have in common, like this:

18: __1__, __2__, __3__, __6__, 9, 18.

24: __1__, __2__, __3__, 4, __6__, 8, 12, 24.

Now, look at the common factors, and find the greatest of the underlined numbers. You would see that 6 is the greatest common factor. This tells you that you can reduce the fraction by 6, like this:

Thus, your reduced fraction is 3/4.

If you have any trouble reducing fractions, or don’t remember how, please review our page on Expanding and Reducing Fractions.