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Bézout coefficients

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

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

Timur

Timur

You can use this calculator to obtain pair of Bézout's coefficients and the general form of Bézout's coefficients as well. 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, the Bézout's identity (aka Bézout's lemma) is the following statement:

Let a and b be integers with 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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Creative Commons Attribution/Share-Alike License 3.0 (Unported) PLANETCALC, Bézout coefficients

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