Extended Euclidean algorithm

Method for computing the relation of two integers with their greatest common divisor

In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that a x + b y = gcd ( a , b) {\displaystyle ax+by=\gcd(a,b)} ; it is generally denoted as xgcd ⁡ ( a , b) {\displaystyle \operatorname {xgcd} (a,b)} . This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor.

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