Congruence Classes

Abstract

In this chapter, we return to the point of view of Example 2.21, looking at congruence modulo n as an equivalence relation. As a byproduct of our discussion, a number of interesting applications will be presented, among which is an alternative, simpler proof of Euler’s theorem. We will also introduce the quotient set $\mathbb Z_n$ and show that it can be furnished with operations of addition and multiplication quite similar to those of $\mathbb Z$. In particular, the case of $\mathbb Z_p$, with p prime, will be crucial to our future discussion of polynomials.