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.