Transcription of The Euclidean Algorithm and Multiplicative Inverses
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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. There s a neat movie demonstration of how the Algorithm worksgeometrically, on theWikipediapage for Euclidean Algorithm .
where q is the quotient, and the remainder r satisfies 0 ≤ r < b. If we clear fractions, this is the equation a = bq +r. We really do know that this last equation is possible: starting with (b)(0), …
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