Residue Classes

Number Theory · Axiom Academy

Understanding how congruence partitions the integers into equivalence classes 1. Congruence as an Equivalence Relation Two integers a and b are congruent modulo n if they differ by a multiple of n. This relationship partitions all integers into equivalence classes. The animation shows how congruence modulo 3 creates three distinct equivalence classes, each containing integers with the same remainder. 2. The Structure of Residue Classes For a given modulus n, there are exactly n residue classes. Each class is represented by a residue (remainder) from 0 to n-1. Watch as we construct the residue classes modulo 5. Each class forms an infinite arithmetic sequence with common difference 5. The set of all residue classes modulo n forms an algebraic structure called the quotient ring Z/nZ . We can add and multiply residue classes, creating a finite ring with n elements.

This is the written version of the interactive lesson above. See the full Number Theory course.