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The Euclidean Algorithm and Multiplicative Inverses

1 The Euclidean Algorithm and Multiplicative InversesLecture notes for Access 2011 The Euclidean Algorithm is a set of instructions for finding the greatest common divisorof any two positive integers. Its original importance was probably as a tool in constructionand measurement; the algebraic problem of findinggcd(a, b) is equivalent to the followinggeometric measuring problem: Given two different rulers, say of lengthsaandb, find athird ruler which is as long as possible, but so that you can still use it as a scale on bothof the longer rulers.

Theorem 2 (Multiplicative Inverse Algorithm). Given two integers 0 < b < a, consider the Euclidean Algorithm equations which yield gcd(a,b) = rj. Rewrite all of these equations ... so that (the residue of) y is the multiplicative inverse of b, mod a. Examples! Example 2. Find integers x and y to satisfy 42823x +6409y = 17.

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