# Congruences with a Prime-power Modulus

• Gareth A. Jones
• J. Mary Jones
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

## Abstract

As we saw in the last chapter, a single congruence mod (n) is equivalent to a set of simultaneous congruences modulo the prime powers pe appearing in the factorisation of n. In this chapter we will therefore study congruences mod (pe), where p is prime. We will first deal with the simplest case e = 1, and then, after a digression concerning primality-testing, we will consider the case e > 1. A good reason for starting with the prime case is that whereas modular addition, subtraction and multiplication behave much the same whether the modulus is prime or composite, division works much more smoothly when it is prime.

## Keywords

Integer Coefficient Modular Arithmetic Elementary Number Theory Linear Congruence Carmichael Number
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