Abstract
Most textbooks on number theory tend to present linear congruences in a more or less theoretical fashion, paying little attention to the mechanical problems inherent in the solution of a congruence with a large modulus such as 73,124x -= 1613 (mod81,763). Some semi-systematic approaches are given, but even the best of such approaches are largely trial and error. The Euclidean algorithm may be used, as in the introductory approach to Diophantine equations found in most books, but this overlooks the use of congruence notation entirely and totally undermines the inherent superiority of the congruence approach to Diophantine equations. The purpose of this note is to present a relatively
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