The Fundamental Theorem of Arithmetic

I have decided to write a series of articles about Cryptography(Which I am always enthusiastic about), starting off with the "Fundamental theorem of Arithmetic". I will try my best to put it in the most simple way.

okay, let's get the ball rolling.

Computers as we all know, will be dumb without mathematics. There is no way a person can understand how computers really work, without the knowledge on mathematics. Cryptography which is the branch of securing information in computers, uses mathematics effectively in order to fulfill its needs. The fundamental theorem of Arithmetic is such a building block which cryptography is depended upon.

The first thing to note is there are two types of numbers, Prime(http://en.wikipedia.org/wiki/Prime_number) and Composite. Each has their own characteristics.

The next step is to understand that any composite number can be built with a prime number adding up several times.

 10= 2 + 2 + 2 + 2 +2
 15=3 + 3 + 3 + 3 + 3
 30= 5+5+5+5+5+5+5

Euclid observed this, and went on building upon that observation.
The next thing he noticed was that a composite number can be expressed with different prime numbers adding up some specific time.It is illustrated below.

30= 2+2+2+2+2+2+2+2+2+2+2+2+2+2+2
30=3+3+3+3+3+3+3+3+3+3
30=5+5+5+5+5+5

With this, we can find the prime factors of that number. i.e 2,3, and 5 are the prime factors of the number 30. Euclid realized that you could multiply these factors a specific amount of times to build that number. In this case each prime factor, once.

30=2*3*5

Take any composite number, this will work. The important thing to notice here is that there is no other way to build 30 by multiplying some other  group of prime numbers together. Every possible number has one and only one possible prime factorization. This gives a unique identification for a composite number. This is known as the  "Fundamental theorem of Arithmetic". Understanding this is very important in terms of cryptography. This idea is one of the main concepts which cryptography has grown on.

I hope you understood what I tried to explain.This doesn't teach you any sort of cryptography, but if you understand this, the rest will really blow your mind.Stay tuned.