Bézout's lemma

Bézout's lemma states that for any two integers, there exists a way to find their greatest common divisor (the largest number that divides both without leaving a remainder) as a linear combination of those two numbers. In other words, you can express their greatest common divisor as \(ax + by\) for some integers \(a\) and \(b\). This means the common divisors are closely related to how these numbers can be combined through addition and subtraction, revealing a fundamental link between divisibility and linear algebra in number theory.