The Fundamental Theorem of Arithmetic (Why Prime Factorization Is Unique)

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either prime or can be written as a unique product of primes. 'Unique' is the powerful word — there's exactly one prime recipe for each number.

Here's what it means and why mathematicians call it fundamental.

What the theorem says

Every integer n > 1 has a prime factorization, and that factorization is unique apart from the order of factors. So 360 is always 2³ × 3² × 5 — never anything else.

Why uniqueness matters

Uniqueness means primes are the genuine 'atoms' of the integers. It guarantees that GCD, LCM, divisor counts and fraction simplification always give consistent answers.

A glimpse of the proof

Existence comes from repeatedly splitting composites into smaller factors until only primes remain. Uniqueness follows from Euclid's lemma: if a prime divides a product, it divides one of the factors.

Where it shows up

It justifies the methods in the Prime Factorization Calculator, the GCD Calculator and the LCM Calculator — all rely on each number having one definite prime signature.