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.

See Also

What is a positive number? What is an integer? What is a factor?