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Bézout's identity - Wikipedia

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In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form az + bt are exactly the multiples of d. Here the greatest common divisor of 0 and 0 is taken to be 0. 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, and this pair is, in the case of integers one of the two pairs such that |x| ≤ |b/d| and |y| ≤ |a/d|; equality occurs only if one of a and b is a multiple of the other. As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as 3 = 15 × (−9) + 69 × 2, with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemm

Bézout's identity - Wikipedia Jump to content From Wikipedia, the free encyclopedia Relating two numbers and their greatest common divisor This article is about Bézout's theorem in arithmetic. For Bézout's theorem in algebraic geometry, see Bézout's theorem . In mathematics , Bézout's identity (also called Bézout's lemma ), named after Étienne Bézout who proved it for polynomials, is a theorem which relates two arbitrary integers with their greatest common divisor . The theorem's statement is as follows: Bézout's identity — Let a and b be integers with greatest common divisor d . Then there ex

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