The Euclidean Algorithm:

1 The greatest common divisor Consider two positive integers a0 > a1. The greatest common divisor of a0 and a1, denoted gcd(a0, a1) is the largest positive integer g such that g|a0 and g|a1, i.e. g divides both a0 and a1. Observation 1: The gcd(a0, a1) always exists. Observation 2 (Euclid): Let a0 = q1a1 + r where 0 ≤ r < a1 (note that this representation is always possible and unique), then gcd(a0, a1) = gcd(a1, r). The proof of this fact consists of showing that d|a0 and d|a1 ⇔ d|a1 and d|r.