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1. Fundamental Theorem Of Arithmetic

   Linked via "smallest counterexample induction"

   Furthermore, this representation is unique, meaning if another factorization exists, say $n = q1^{b1} q2^{b2} \cdots qm^{bm}$ where $qi$ are primes, then $k=m$, the set of primes $\{pi\}$ is identical to $\{qi\}$, and the corresponding exponents must be equal ($ai = b_i$ after appropriate reordering).
   The existence part of the theorem is sometimes proven using the principle of smallest counterexample induction, while the uniqueness part relies fundamentally on Euclid's Lemma…