The fundamental theorem of arithmetic

Statement. Every integer $n\ge2$ can be written uniquely as a product of primes (up to order).

Existence. By strong induction: $n$ has a prime divisor $p$ (previous exercise), $n=p\cdot m$ with $m<n$; if $m=1$ done, else $m$ factors by induction.

Uniqueness. Suppose $n=p_1\cdots p_r=q_1\cdots q_s$. Since $p_1\mid q_1\cdots q_s$, Euclid's lemma gives $p_1=q_j$ for some $j$. Cancel and repeat by induction.

Conclusion. The prime factorization exists and is unique.