Proof

Find the gcd of a and b using Euclid's algorithm . When after some steps using this algorithm

a' = xa + yb

and

b' = ua + vb

for certain integers x, y, u and v, then after the next step

a' = ua + vb

and

b' = (xa + yb) - q(ua + vb)

= (x - qu)a + (y - qv)b,

where q is the quotient of a' and b'. Since Euclid's algorithm will eventually return a' as the gcd of a and b, the extended Euclidean algorithm will give x and y with gcd(a,b) = xa + yb.