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Bezout's Identity

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

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Bezout's Identity -

Tag(s) : Arithmetics

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# Bezout's Identity

## Bezout Identity Calculator

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

### What is Bezout Identity?

The Bachet-Bezout identity is defined as : if $a$ and $b$ are two integers and $d$ is their GCD (greatest common divisor), 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 : $$12 \times -2 + 30 \times 1 = 6$$

The dCode Bezout coefficients calculator gives only one solution.

### What are Bezout coefficients?

The Bézouts coefficients are the values $u$ and $v$.

### 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 (and also the GCD).

### 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)

## Source code

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