Prime Factorization

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

Example 1: What are the prime factors of 12 ?

12 = 2 × 2 × 3

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

Factor Tree
And 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.

http://en.wikipedia.org/wiki/Prime_factor

Perfect square numbers can be recognized by the fact that all of their prime factors have even multiplicities. For example, the number 144 (the square of 12) has the prime factors

These can be rearranged to make the pattern more visible:

Because every prime factor appears an even number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on.

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 basic building blocks of all numbers.

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

Cryptography is the study of secret codes. Prime Factorization is very 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.

If you want to know more, the subject is "encryption" or "cryptography".
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.

In fact this idea is so important it is called the Fundamental Theorem of Arithmetic.

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