Divisibility Theory

Intro to Proofs · Axiom Academy

The formal foundations of divisibility: definitions, fundamental properties, and the division algorithm with proof of uniqueness. Key points about the notation: is a statement (true or false), not an operation means b = ak for some integer k means no such integer k exists We require to avoid division-by-zero issues " a divides b " means b is exactly a copies, with nothing left over Since , there exists such that b = ak_1 Since , there exists such that c = bk_2 3. Linear Combination Property Since , there exists such that c = ak_2 This theorem has two parts: existence and uniqueness . We'll focus on proving uniqueness — that there is only one valid pair (q, r) — in the next step. Two valid remainders differ by less than b : |r_1 - r_2| < b . That difference is also a multiple of b — the only one small enough is 0 . You've built divisibility from its formal definition up through the division algorithm and its uniqueness proof — the bedrock of number theory. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.