# Recursive Extended Euclidean Algorithm

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The greatest common divisor of positive integers , , …, is the largest integer such that divides each of the integers. There always exist integers , , …, such that .

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Contributed by: Roger B. Kirchner (March 2011)

Open content licensed under CC BY-NC-SA

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The correctness of the recursive definitions in the source code can be argued by observing 1) they terminate, 2) they return the correct result when they terminate, and 3) they return the correct result whenever a recursive call returns the correct result. The function in the program for is ; it is defined there for reference. Only is used, however. It has a list of positive integers and a recursion level for inputs and returns the list that is displayed.

A classic exposition of these ideas is found in "1.7 The Euclidean Algorithm", in Birkhoff and MacLane, which can be read online.

The recursive algorithm we have used (for ) is essentially described in Euclidean algorithm, under "Extended Euclidean Algorithm".

G. Birkhoff and S. MacLane, *A Survey of Modern Algebra,* 4th ed., New York: MacMillan Publishing Co., 1977.

K. H. Rosen, *Elementary Number Theory and its Applications, *4th ed., Reading, MA: Addison–Wesley, 2000.

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