# Prime Factorization

## Prime Numbers

A Prime Number is:

a whole number above 1 that **cannot** be made by multiplying other whole numbers

When it **can** be made by multiplying other whole numbers it is a **Composite Number**, like this:

(The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19 and 23, and we have a prime number chart if you need more.)

## Factors

"Factors" are the numbers you multiply together to get another number:

## Prime Factorization

"Prime Factorization" is finding **which prime numbers** multiply together to make the original number.

Here are some examples:

### Example: 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

Yes, it divided exactly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let's try 2 again:

6 ÷ 2 = 3

Yes, that worked also. And 3 **is** a prime number, so we have the answer:

**12 = 2 × 2 × 3**

As you can see, **every factor** is a **prime number**, so the answer is right.

It is neater to show repeated numbers using exponents:

- Without exponents:
**2 × 2 × 3** - With exponents:
**2**^{2}× 3

### Example: What is the prime factorization of 147 ?

Can we divide 147 exactly by 2?

147 ÷ 2 = 73½

No we can't. The answer should be a whole number, and 73½ is not.

Let's try the next prime number, 3:

147 ÷ 3 = 49

That worked, now try factoring 49.

The next prime, 5, does not work. But 7 does, so we get:

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 × 7 ^{2}**

### Example: What is the prime factorization of 17 ?

Hang on ... **17 is a Prime Number**.

So that is as far as we can go.

**17 = 17**

## Another Method

We just did factorization by starting at the smallest prime and working upwards.

But sometimes it is easier to break a number down into **any factors** we can ... then work those factor down to primes.

### Example: What are the prime factors of 90 ?

Break 90 into 9 × 10

- The prime factors of 9 are
**3 and 3** - The prime factors of 10 are
**2 and 5**

So the prime factors of 90 are **3, 3, 2 and 5**

**90 = 2 × 3 ^{2} × 5**

## Factor Tree

A "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

Now we continue and factor 8 into **4 × 2**

Then 4 into **2 × 2**

And lastly 6 into **3 × 2**

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

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

**48 = 2 ^{4} × 3**

## 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.

It is like the Prime Numbers are the |

This idea can be very useful when working with big numbers, such as in Cryptography.

## Cryptography

Cryptography is the study of secret codes. Prime Factorization is important to people who try to make (or break) secret codes based on numbers.

That is because factoring very large numbers is very hard, and can take computers a long time to do.

## Unique

And here is another thing:

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

### Example: the prime factors of 330 are 2, 3, 5 and 11

330 = 2 × 3 × 5 × 11

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

This idea is so important it is called the Fundamental Theorem of Arithmetic.

## Prime Factorization Tool

OK, we have one more method ... use our Prime Factorization Tool that can work out the prime factors for numbers up to 9007199254740991.