The extended Euclidean algorithm finds integers x and y such that a × x + b × y = gcd(a, b). This is called Bezout identity.
Example: gcd(12, 18) = 6. You can write 6 = 12 × (-1) + 18 × 1. Here x = -1, y = 1.
This is useful for modular inverses and solving linear Diophantine equations. You will not see it often, but when you need it, you need it.