Extended Euclidean Algorithm

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The greatest common divisor of two integers and
can be found by the Euclidean algorithm by successive repeated application of the division algorithm. The extended Euclidean algorithm not only computes
but also returns the numbers
and
such that
. The remainder
of the
step in the Euclidean algorithm can be expressed in the form
, where
and
can be determined from the corresponding quotient
and the values
, or
two rows above them using the relations
and
, respectively. This forward method requires no back substitutions and reduces the amount of computation involved in finding the coefficients
and
of the linear combination.
Contributed by: Štefan Porubský (March 2011)
Based on a program by: Michael Trott
Open content licensed under CC BY-NC-SA
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The result saying that the greatest common divisor of any two integers and
can be written as their linear combination is also known as the Bachet–Bézout theorem (actually, Bézout formulated it for polynomials).
For more information, see the Wikipedia entry for Extended Euclidean algorithm.
The underlying Mathematica code is an adaptation of the code used in the Demonstration Euclidean Algorithm Steps by Michael Trott.
(The author was supported by project 1ET200300529 of the Information Society of the National Research Program of the Czech Republic and by the Institutional Research Plan AV0Z10300504; the Demonstration was submitted 2008-06-20, revised 2010-03-13.)
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