# Bézout coefficients

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

### This page exists due to the efforts of the following people:

#### Karen Luckhurst

Created: 2020-02-12 07:03:08, Last updated: 2020-12-29 11:59:14

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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

#### 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

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

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PLANETCALC, Bézout coefficients