dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day! You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!

This page is using the new English version of dCode, please make comments !

Tool to compute Bezout coefficients. The Bezout Identity proves that it exists solutions to the equation a.u + b.v = PGCD(a,b).

Answers to Questions

What is Bezout Identity?

The Bezout identity says that if \( a \) and \( b \) are two integers and \( d \) is their GCD, then it exists \( u \) and \( v \), two integers such as \( au + bv = d \).

Example: \( a=12 \) and \( b=30 \), gcd \( (12, 30) = 6 \). There are multiple solutions to \( u \) and \( v \) such as \( 12u + 30v = 6 \), such as : $$ 12 \times -2 + 30 \times 1 = 6 $$

How to calculate values for Bézout Identity?

The dCode program uses the extended GCD algorithm. \( a \) and \( b \) are two non-zero positive integers.

The algorithm of dCode consists of a sequence of Euclidean divisions for finding the Bezout coefficients.

How to code Bézout Identity in pseudo-code?

A source code for the identity of Bezout would be similar to this pseudo-code:

Initialization r = a, r' = b, u = 1, v = 0, u' = 0 and v' = 1 While (r' != 0) q = (int) r/r' rs = r, us = u, vs = v, r = r', u = u', v = v', r' = rs - q*r', u' = us - q*u', v' = vs - q*v' End While Return (r, u, v)

Ask a new question

Source code

dCode retains ownership of the source code of the script Bezout's Identity. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Bezout's Identity script for offline use, for you, your company or association, see you on contact page !

dCode uses cookies to customize the site content, analyze user behavior and adapt dCode to your use. Some data is stored and collected for advertising purposes and may be shared with our partners. OK