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.

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