Applications of the Fundamental Theorem of Arithmetic - 2

Finding all the factors of a number.

Hopefully it is fairly obvious that the factors of 12 are 1, 2, 3, 4, 6 and 12.
Personlly, I like to have them written down in pairs, as it helps to know that you've found them all:

12
1     12
2     6
3     4

Using the Fundamental Theorem of Arithmetic:

12 = 2²×3

or, in expanded form:

12 = 2×2×3

How does this help? Well, 1 is a factor of every number, so we include it anyway. Every other factor of 12 can be made with every different combination of its prime factors:

1 - factor of every number

2 - 2

3 - 3

4 - 2×2

6 - 2×3

12 - 2×2×3

To find all the factors of 72:

72 = 2³×3²

or, in expanded form:

72 = 2×2×2×3×3

So, the possible factors are:

1 - factor of every number

2 - 2

3 - 3

4 - 2×2

6 - 2×3

8 - 2×2×2

9 - 3×3

12 - 2×2×3

18 - 2×3×3

24 - 2×2×2×3

36 - 2×2×3×3

72 - 2×2×2×3×3

The interesting thing about this method is that all of the factors are written as products of prime numbers too.

Can you use this method to find all the factors of 54?

What about 105?

Or 120?