# What Is the Fundamental Theorem of Arithmetic? (Mathematics Lesson)

# 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.#### An Example Question

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

What is 6 as a product of prime numbers?#### An Example 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.

##### Interactive Test

**show**

Here's a second test on the fundamental theorem of arithmetic.

Here's a third test on the fundamental theorem of arithmetic.

##### Note

# 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.