Bézout's identity

Bézout's identity (also called Bézout's lemma) is a theorem in the elementary theory of numbers: let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x and y such that

In addition,

The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm. If both a and b are nonzero, the extended Euclidean algorithm produces one of the two pairs such that and

Bézout's lemma is true in any principal ideal domain, but there are integral domains in which it is not true.

Structure of solutions

When one pair of Bézout coefficients (x, y) has been computed (e.g., using extended Euclidean algorithm), all pairs can be represented in the form

where k is an arbitrary integer and the fractions simplify to integers.