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For any integers a and b, there exist integers s and t such that as + bt = gcd(a,b). The proof of this identity, as well as a method for finding s and t, is contained in the Extended Euclidean Algorithm. Bezout's Identity is useful for solving equations in modular rings (and, as a common special case, finding modular inverses) and solving Diophantine Equations.

Bezout's Identity is named for the 18th century French mathematician Etienne Bezout.