On the Extended Euclidean Algorithm

Abstract

The algorithm terminates after a finite number of iterations, since b is replaced in each iteration by the remainder r = a mod b, which is a nonnegative integer that is strictly smaller than b. Therefore, the algorithm terminates after at most b iterations. We show now that the algorithm is correct. We denote by 〈a, b〉 the set {ax + by | x, y ∈ Z}; this set is called the ideal generated by a and b in the ring of integers. Notice that if 〈a, b〉 contains the integers c and d, then 〈c, d〉 is a subset of 〈a, b〉. Lemma 1 If b 6= 0, then 〈a, b〉 = 〈b, a mod b〉. Proof. The ideal 〈a, b〉 contains the remainder r = a mod b, since r = a− qb with q = ba/bc. Thus, if b 6= 0 then the ideal 〈b, a mod b〉 is a subset of 〈a, b〉.