## The Lesson

The fundamental theorem of arithmetic states that:
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? This process is prime factorisation, as every number can be written as a product of prime factors.

## 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.