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Mathematics LibreTexts

3: Congruences


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A congruence is nothing more than a statement about divisibility. The theory of congruences was introduced by Carl Friedreich Gauss. Gauss contributed to the basic ideas of congruences and proved several theorems related to this theory. We start by introducing congruences and their properties. We proceed to prove theorems about the residue system in connection with the Euler \(\phi\)-function. We then present solutions to linear congruences which will serve as an introduction to the Chinese remainder theorem. We present finally important congruence theorems derived by Wilson, Fermat and Euler.


  • Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.