What is bézout's identity?

Bézout's identity states that any two integers a and b which are not both equal to zero have a greatest common divisor (gcd) which can be expressed as a linear combination of a and b. Specifically, this means that there exist integer values x and y such that gcd(a,b) = ax + by. This identity is named after the French mathematician Étienne Bézout who first discovered it in the mid-18th century. It is a fundamental concept in number theory and is used in various mathematical applications, such as modular arithmetic and cryptography.