Factors

Continue to explore the question of divisibility by asking: What numbers is 6 divisible by? Questions related to divisibility are only interested in positive integers, so you really only have six possible numbers: 1, 2, 3, 4, 5, and 6. You can test to see which numbers 6 is divisible by:

6 ÷ 1 = 6 Any number divided by 1 equals itself, so an integer divided by 1 will be an integer.
Note that these form a pair.
Theses are not integers, so 6 is NOT divisible by 4 or by 5.
6 ÷ 6 = 1 Any number divided by itself equals 1, so an integer is always divisible by itself.

So 6 is divisible by 1, 2, 3, and 6. That means that 1, 2, 3, and 6 are factors of 6. There are a variety of ways you might see this relationship expressed on the GRE:

2 is a factor of 6 6 is a multiple of 2
2 is a divisor of 6 6 is divisible by 2
2 divides 6 2 goes into 6

Sometimes it will be necessary to find the factors of a number in order to answer a question. An easy way to find all the factors of a small number is to use factor pairs. Factor pairs for any integer are the pairs of factors that, when multiplied together, yield that integer.

Here’s a step-by-step way to find all the factors of the number 60 using a factor pairs table:

  1. Make a table with two columns labeled “Small” and “Large.”
  2. Start with 1 in the small column and 60 in the large column. (The first set of factor pairs will always be 1 and the number itself.)
  3. The next number after 1 is 2. If 2 is a factor of 60, then write “2” underneath the “1” in your table. It is, so divide 60 by 2 to find the factor pair: 60 ÷ 2 = 30. Write “30” in the large column.
  4. The next number after 2 is 3. Repeat this process until the numbers in the small and the large columns run into each other. In this case, 6 and 10 are a factor pair. But 7, 8, and 9 are not factors of 60, and the next number after 9 is 10, which appears in the large column, so you can stop.

    Small Large
    1 60
    2 30
    3 20
    4 15
    5 12
    6 10

The advantage of using this method, as opposed to thinking of factors and listing them out, is that this is an organized, methodical approach that makes it easier to find every factor of a number quickly. Let’s practice. (This is also a good opportunity to practice your long division.)

Check Your Skills

  1. Find all the factors of 90.

  2. Find all the factors of 72.

  3. Find all the factors of 105.

  4. Find all the factors of 120.