Prime Factorization

A Prime Number is a whole number, greater than 1, that can be evenly divided only by 1 or itself.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17.

Factors
Factors are the numbers you multiply together to get another number. For example, 2 x 3 = 6. 2 and 3 are factors of 6 because when you multiply them you get 6. 1 and 6 are also factors of 6 because when you multiply 1 and 6, you get 6.


Prime Factorization
Prime Factorization is finding which prime numbers you need to multiply together to get the original number. Think factor tree!

Example 1

What are the prime factors of 12?

It is best to start working from the smallest prime number, which is 2, so let’s check:

12 ÷ 2 = 6

But 6 is not a prime number, so we need to factor it further:

6 ÷ 2 = 3

And 3 is a prime number, so:

12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right - the prime factorization of 12 is 2 × 2 × 3, which can also be written as 22 × 3

Example 2

What is the prime factorization of 147?

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3:

147 ÷ 3 = 49

Then we try factoring 49, and find that 7 is the smallest prime number that works:

49 ÷ 7 = 7

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7 = 3 × 72


Why?

A prime number can only be divided by 1 or itself, so it cannot be factored any further!

Every other number can be broken down into prime number factors.

So, in a way, the prime numbers are the building blocks of all other numbers.