Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 4 / 2E / a / Solution

Every integer $n>1$ is a product of primes, uniquely up to order. Existence follows by strong induction: if $n$ is not prime, factor it into smaller positive integers and apply induction. For uniqueness, ifEuclid's lemma makes $p_1$ divide some $q_j$, hence equal it. Cancel that prime and induct on the number of factors.