Any positive integer greater than 1 is either a prime number, or a unique product of prime numbers.

In this definition,
- A
*"positive integer greater than 1"*means 2, 3, 4, 5, 6 etc... - A
*"prime number"*is a number that can be divided by only itself and 1 (for example, 5 can only be divided exactly by 1 and 5 itself). - A
*"product of prime numbers"*means two or more prime numbers multiplied together. (**Note:**These are called composite numbers).

## What Does the Fundamental Theorem of Arithmetic Mean?

The fundamental theorem of arithmetic means that all numbers are either prime numbers or can be found by multiplying prime numbers together:## Prime Numbers and Composite Numbers

All positive integers greater than 1 are either a prime number or a composite number.## Composite Numbers As Products of Prime Numbers

By the fundamental theorem of arithmetic, all composite numbers must be a product of prime numbers.## Question

What is 4 as a product of prime numbers?## Question

What is 6 as a product of prime numbers?## Question

What is 8 as a product of prime numbers?## The Uniqueness of Prime Factors

Not only can any number be written as a product of prime numbers, but the prime factors are unique. 8, for example, can only be found by 2 × 2 × 2. No other group of prime numbers can be multiplied together to find 8.## What Is a Prime Number?

A prime number is a number that can be divided exactly by only itself and 1. For example, 5 is a prime number. It can only be divided by 1 and 5 itself.## What Is a Composite Number?

A composite number is a number with at least one other factor besides itself and 1. A composite number is a number that is not a prime number. For example, 4 is a composite number. It can not only be divided exactly by 1 and 4, but also by 2. That is, it has one other factor besides itself and 1.## Who Discovered the Fundamental Theorem of Arithmetic?

The famous ancient Greek mathematician Euclid first stated the theorem in his famous*Elements*book, which is perhaps the most read and longest running textbooks of all time.

## Why Is It Useful?

Representing numbers as prime factors is very important in encryption - encoding messages so only those authorized can read them. Any number can be written as a product of two prime numbers. But it is very difficult to work out which two prime numbers, especially if it is a very large number. If decoding a message requires a large number to be broken down into two prime numbers, it is beyond current computers to do this in a reasonable time.## You might also like...

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