The Euclidean Algorithm
نویسنده
چکیده
Euclid’s algorithm gives the greatest common divisor (gcd) of two integers, gcd(a, b) = max{d ∈ Z | d|a, d|b} If for simplicity we define gcd(0, 0) = 0, we have a function gcd : Z× Z −→ N with the following properties: Lemma 1 For any a, b, c, q ∈ Z we have: (i) gcd(a, b) = gcd(b, a). (ii) gcd(a,−b) = gcd(a, b). (iii) gcd(a, 0) = |a|. (iv) gcd(a− qb, b) = gcd(a, b). Proof. Trivial; for (iv) use the equivalence d|a, b⇐⇒ d|a− qb, b. 3
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