Bézout coefficients

This online calculator computes Bézout's coefficients for two given integers, and represents them in the general form

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Karen Luckhurst

Created: 2020-02-12 07:03:08, Last updated: 2020-12-29 11:59:14
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This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/8586/. Also, please do not modify any references to the original work (if any) contained in this content.

You can use this calculator to obtain a pair of Bézout's coefficients as well as the general form of the coefficients. Some theory can be found below the calculator

PLANETCALC, Bézout coefficients

Bézout coefficients

Bézout coefficients

First coefficient
Second coefficient
General form

Bézout's identity and Bézout's coefficients

To recap, Bézout's identity (aka Bézout's lemma) is the following statement:

Let a and b be integers with the greatest common divisor d. Then, there exist integers x and y such that ax + by = d. More generally, the integers of the form ax + by are exactly the multiples of d.

If d is the greatest common divisor of integers a and b, and x, y is any pair of Bézout's coefficients, the general form of Bézout's coefficients is

\left(x+k\frac{b}{\gcd(a,b)},\ y-k\frac{a}{\gcd(a,b)}\right)

and the general form of Bézout's identity is


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PLANETCALC, Bézout coefficients