Document Type


Date of Award



College of Science, Engineering, and Technology (COSET)

Degree Name

MS in Mathematics

First Advisor

Professor Robert Nehs


All variables are integers, n > 1 and let 7l be the set of integers. Two integers a and b are said to be congruent modulo n if and only if n divides b - a; equivalently, b = a + nt for some integer t. Congruence modulo n is an equivalence relation on Z and thus partitions Z into a set of equivalence classes a, called residue classes. The set of residue classes is tln = {o,1,2,. . ·,n-1}, the integers modulo n. Addition and multiplication of integers is used to define the corresponding operations on the residue classes, written: a + b and a . b. Accordingly, tln with these operations forms an algebraic system called a ring, The residue class 1 is the identity for multiplication because a • 1 = 1 . a = a. A residue class a in tln is invertible if and only if there is a residue class b in tln such that a • b = b . a = 1. In this case, b is called the 'inverse of a and is denoted a-I. When a-􁪽 = a, a is said to be self-invertible. Thus a is self-invertible if and only if a . a = 1.