Applications of the Fundamental Theorem of Arithmetic - 3

The number of factors

Not only does the Fundamental Theorem of Arithmetic help us to find all the factors of a number, it is a quick way to know how many factors a number has in total.

Take the number 24:

1     24

2     12

3     8

4     6

So 24 has 8 factors.

Written as prime factors:

24 = 2³×3

If we take the powers: 3 and 1

Increase them each by one: 4 and 2

Then multiply them together: 4×2 = 8

So 24 has 8 factors.

Try another number: 180

180 = 2²×3²×5

The powers are 2, 2 and 1.

Increase them each by 1 to get 3, 3 and 2.

Multiply them together: 3×3×2 = 18

So 180 should have 18 factors.

This can be checked by working them out:

180

1     180

2     90

3     30

4     45

5     36

6     30

9     20

10     18

12     15

Can you find out how many factors 48 has?

What about 216?

Or 1260?

Can you list all the factors?