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Mathematical Sciences Colloquium

Peter Hamburger, Western Kentucky, and IPFW Emeritus

*An Alternative Proof of Bézout's Theorem*, joint work with with G. Petruska

- In this talk an alternative proof of Bézout's Theorem will be given. Bézout's Theorem states that if gcd(
*a*,*b*)=*d*is the greatest common divisor of two integers*a*and*b*, then there are integers*s*and*t*such that*sa*+*tb*=*d*. The integers*s*and*t*are called Bézout's coefficients. This proof does not use the Euclidean Algorithm, or more precisely it does not use the Extended Euclidean Algorithm. The algorithm which is used in the proof gives both the greatest common divisor of two integers*a*and*b*and the Bézout's coefficients simultaneously. Then we modify the algorithm to simplify it. The modified algorithm contains one Division Algorithm, (the same as the first step of the Euclidean Algorithm), then defines an arithmetic progression that leads to the greatest common divisor and the first Bézout's coefficient at the same time; to find the second Bézout's coefficient an additional division is needed.

The talk is suitable for students who are in MA 175/275 Discrete Mathematics.

Nov. 20, 2012