Modular Arithmetic

Definition (Modular Arithmetic) Let   be a complete set of equivalence class representatives on for congruence modulo then and define the operation on by Arithmetic using this operation is often referred to as modular arithmetic.

    The operation is a well-defined binary operation; meaning, if and in then  



which follows from

Proposition (Modular Arithmetic) Let be a positive integer, then is associative and commutative, each element has an inverse, and is the identity element.

    Proof. By the definition of and the use of associativity in the integers, it follows that for any   and The element is the identity because for any For every element of there is an inverse because and indeed   Commutativity follows since for

Example (Modular Arithmetic) The arithmetic table for is:

Proposition (Modular Exponentiation) If and then

    Proof. Because we have by definition, and since
    


we see that whence Therefore,

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Cite this as:
Modular Arithmetic
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/modular-arithmetic.html