fundamental theorem of arithmetic

Think of primes as a set of standard building blocks. The fundamental theorem of arithmetic promises two things at once: every whole number above 1 can be built from these blocks, and there is essentially only one way to build each number. No matter how you go about factoring, you cannot reach two genuinely different prime recipes for the same number.

Stated carefully: every integer greater than 1 is either prime or can be written as a product of primes, and that product is unique up to the order in which the factors are listed. So 60 = 2 times 2 times 3 times 5, and any correct factorization will use exactly two 2s, one 3, and one 5 — never anything else.

This is the bedrock under most of elementary number theory. Greatest common divisors, least common multiples, and the behavior of fractions in lowest terms all rest on it. The ordering caveat is the only flexibility: 2 times 3 and 3 times 2 count as the same factorization. The theorem is also exactly why 1 is not called prime — admitting 1 would let you pad any product with extra 1s and destroy uniqueness.