Bézout's identity

Bézout's identity (also called Bézout's lemma) is a theorem in elementary number theory: 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 ${\displaystyle |x|\leq \left|{\frac {b}{d}}\right|}$ and ${\displaystyle |y|\leq \left|{\frac {a}{d}}\right|}$ (equality may occur only if one of a and b is a multiple of the other).

Many theorems of elementary theory of numbers, such as Euclid's lemma or Chinese remainder theorem, result from Bézout's identity.

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