# Counting factors

Counting factors is not a bad idea when you are looking for the factors of a number.

You may need this information if you do not want to miss any factors.

To count factors, first we need to get the prime factorization of the number

Example #1:

How many factors does 8 have?

8 = 2 × 2 × 2 = 23

The factors of 8 are 1, 2, 4, 8. There are 4 factors

Looking at 23, we notice that if we add 1 to the exponent, we get the 4

However, just one case is not enough to conclude that when counting factors the number of factors is whatever the exponent is plus 1

Let's look at more examples

Example #2:

How many factors does 25 have?

25 = 5 × 5 = 52

The factors of 25 are 1, 5, and 25. 25 has 3 factors

Again, to get the 3, just add 1 to the exponent of 2

Example #3:

How many factors does 72 have?

72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = 23 × 32

The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. There are 12 factors

How do we get the 12? By adding 1 to each exponent and then multiply

(3 + 1) × (2 + 1) = 4 × 3 = 12

So far it seems like adding 1 is a good strategy when counting factors

Another way to see all the factors of 72 are shown below:

20 × 30 = 1 × 1 = 1

20 × 31 = 1 × 3 = 3

20 × 32 = 1 × 9 = 9

21 × 30 = 2 × 1 = 2

21 × 31 = 2 × 3 = 6

21 × 32 = 2 × 9 = 18

22 × 30 = 4 × 1 = 4

22 × 31 = 4 × 3 = 12

22 × 32 = 4 × 9 = 36

23 × 30 = 8 × 1 = 8

23 × 31 = 8 × 3 = 24

23 × 32 = 8 × 9 = 72

As you can see there are 4 choices for the exponents of 2: 0, 1, 2, 3.

And 4 choices for the exponents of 3: 0, 1, 2

4 choices × 3 choices = 12 choices and this is equal to 12 factors

Example #4:

How many factors 12600 have?

When counting factors for big numbers, it may be useful to make a factor tree

Pull out all the prime numbers from the tree and multiply the numbers. This is your prime factorization.

2 × 2 × 2 × 3 × 3 × 5 × 5 × 7

23 × 32 × 52 × 71

Add 1 to each exponent and multiply:

(3 + 1) × (2 + 1) × (2 + 1) × (1 + 1)

4 × 3 × 3 × 2

12 × 3 × 2

36 × 2 = 72

12600 has 72 factors.

Take the counting factors quiz below to see how well you understand this lesson.

## Recent Articles

1. ### Average age word problem

Aug 09, 16 01:40 PM

Average age of Dipu and Apu is 22 years. Average age of Dipu and Tipu is 24 years. Age of Dipu is 21 years. What are the ages of Apu, and tipu ? Let