Elementary Number Theory pp 37-63 | Cite as

# Congruences

- 4.4k Downloads

## Abstract

In this chapter, we will study modular arithmetic, that is, the arithmetic of congruence classes, where we simplify number-theoretic problems by replacing each integer with its remainder when divided by some fixed positive integer _{n}. This has the effect of replacing the infinite number system ℤ with a number system ℤ_{n} which contains only n elements. We find that we can add, subtract and multiply the elements of ℤ_{n} (just as in ℤ), though there are some difficulties with division. Thus ℤ_{n} inherits many of the properties of ℤ, but being finite it is often easier to work with. After a thorough study of linear congruences (the analogues in ℤ_{n} of the equation *ax* = *b*), we will consider simultaneous linear congruences, where the Chinese Remainder Theorem and its generalisations play a major role.

## Preview

Unable to display preview. Download preview PDF.