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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

How to calculate values for Bézout 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 \( au + bv = d \).

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

Consider \( a=12 \) and \( b=30 \), you get gcd \( (12, 30) = 6 \), you look for \( u \) and \( v \) such as \( 12u + 30v = 6 \). There are multiple solutions, for example : $$ 12 \times -2 + 30 \times 1 = 6 $$

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

How to code Bézout Identity in pseudo-code?

Initialization r = a, r' = b, u = 1, v = 0, u' = 0 et 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)

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Source code

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