In mathematics, congruence modulo (referred to as well modular arithmetic) is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus, see https://en.wikipedia.org/wiki/Modular_arithmetic. The idea is simply to divide by and then take the reminder.
Examples
Definition
Let be a positive integer. Then two integers and are said to be congruent mod , written as if
Question
Let be a positive integer. Show that the relation of congruence mod is an equivalence relation.
Solution
Note that
Thus,
(i) Reflexive:
(ii) Symmetric: Suppose
(iii) Transitive: Suppose and , then and
So and hence
Quiz
Show further that there are equivalence classes of the form
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Congruence Modulo n and Equivalence Relations
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