How to Generate a Number’s Factors

When one number is divisible by a second number, that second number is a factor of the first. For example, 10 is divisible by 2, so 2 is a factor of 10.

A good way to find all the factors of a number is by finding all of that number’s factor pairs. A factor pair of a number is any pair of two numbers that, when multiplied together, equal that number. For example, 30 has four factor pairs —1 x 30, 2 x 15, 3 x 10, and 5 x 6 — because

1 x 30 = 30

2 x 15 = 30

3 x 10 = 30

5 x 6 = 30

Here’s how to find all the factors pairs of a number:

1. Begin the list with 1 times the number itself.

2. Try to find a factor pair that includes 2 — that is, see whether the number is divisible by 2.

   If it is, list the factor pair that includes 2.

3. Test the number 3 in the same way.

4. Continue testing numbers until you find no more factor pairs.

Sample question

1. Write down all the factor pairs of 18.

   1 x 18, 2 x 9, 3 x 6. According to Step 1, begin with 1 x 18:

   1 x 18

   The number 18 is even, so it’s divisible by 2. And 18 / 2 = 9, so the next factor pair is 2 x 9:

   1 x 18

   2 x 9

   The digital root of 18 is 9 (because 1 + 8 = 9), so 18 is divisible by 3. And 18 / 3 = 6, so the next factor pair is 3 x 6:

   1 x 18

   2 x 9

   3 x 6

   The number 18 isn’t divisible by 4, because 18 / 4 = 4 r 2. And 18 isn’t divisible by 5, because it doesn’t end with 5 or 0. This list of factor pairs is complete because there are no more numbers between 3 and 6, the last factor pair on the list.

Practice questions

1. Find all the factor pairs of 12.

2. Write down all the factor pairs of 28.

3. Figure out all the factor pairs of 40.

4. Find all the factor pairs of 66.

Following are the answers to the practice questions:

1. The factor pairs of 12 are 1 x 12, 2 x 6, and 3 x 4.

   The first factor pair is 1 x 12. And 12 is divisible by 2 (12 / 2 = 6), so the next factor pair is 2 x 6. Because 12 is divisible by 3 (12 / 3 = 4), the next factor pair is 3 x 4.

2. The factor pairs of 28 are 1 x 28, 2 x 14, and 4 x 7.

   The first factor pair is 1 x 28. And 28 is divisible by 2 (28 / 2 = 14), so the next factor pair is 2 x 14. Although 28 isn’t divisible by 3, it’s divisible by 4 (28 / 4 = 7), so the next factor pair is 4 x 7. Finally, 28 isn’t divisible by 5 or 6.

3. The factor pairs of 40 are 1 x 40, 2 x 20, 4 x 8, and 5 x 8.

   The first factor pair is 1 x 40. Because 40 is divisible by 2 (40 / 2 = 20), the next factor pair is 2 x 20. Although 40 isn’t divisible by 3, it’s divisible by 4 (40 / 4 = 10), so the next factor pair is 4 x 10.

   And 40 is divisible by 5 (40 / 5 = 8), so the next factor pair is 5 x 8. Finally, 40 isn’t divisible by 6 or 7.

4. The factor pairs of 66 are 1 x 66, 2 x 33, 3 x 22, and 6 x 11.

   The first factor pair is 1 x 66. The number 66 is divisible by 2 (66 / 2 = 33), so the next factor pair is 2 x 33. It’s also divisible by 3 (66 / 3 = 22), so the next factor pair is 3 x 22.

   Although 66 isn’t divisible by 4 or 5, it’s divisible by 6 (66 / 6 = 11), so the next factor pair is 6 x 11. Finally, 66 isn’t divisible by 7, 8, 9, or 10.