# Prime Factorization

## Prime Numbers

A Prime Number is a whole number greater than 1 that can not be made by multiplying other whole numbers.

Example

2, 3, 5, 7, 11, 13, 17, 19, 23

The number that we can make by multiplying other whole numbers is a Composite Number.

## Factors

Factors are the numbers you multiply together to get another number

4 × 2 = 8

From example above, the factors are 4 and 2

## Prime Factorization

Prime Factorization is a method to find which prime numbers multiply together to make the original number.

Example

What are the prime factors of 28 ?

Start working from the smallest prime number, which is 2,

so let’s check:

28 ÷ 2 = 12

It divided exactly by 2.

But 12 is not a prime number, so we need to go further. Let’s try 2 again:

14 ÷ 2 = 7

Yes, that worked also. And 7 is a prime number, so we have the answer:

28 = 2 × 2 × 7

Every factor is a prime number, so the answer must be right.

Note: 28 = 2 × 2 × 7 can also be written using exponents as 28 = 22 × 7

## Another Method

Sometimes it is easier to break a number down into any factors you can, then work those factor down to primes.

Example

What are the prime factors of 100 ?

Break 100 into 4 × 25

The prime factors of 4 are 2 and 2

The prime factors of 25 are 5 and 5

So the prime factors of 90 are 2, 2, 5 and 5

## Factor Tree

Factor Tree can help find any factors of the number, then the factors of those numbers, etc, until we can’t factor any more.

Example

48

48 = 8 × 6,

so we write down 8 and 6 below 48

factor 8 into 4 × 2

Then 6 into 3 × 2

lastly 4 into 2 × 2

We can’t factor any more, so we have found the prime factors.

Which reveals that 48 = 2 × 2 × 2 × 2 × 3

48 = 24 × 3 (using exponents)

## Why find Prime Factors?

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

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

## Unique Prime Factors

There is only one unique set of prime factors for any number.

330 = 2 × 3 × 5 × 11

The prime factors of 330 are 2, 3, 5 and 11

There is no other possible set of prime numbers that can be multiplied to make 330.

This idea is important and called the Fundamental Theorem of Arithmetic.