# What Is the Fundamental Theorem of Arithmetic?

## What Is the Fundamental Theorem of Arithmetic?

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.