Introduction

Numbers are often classified using a very simple test: what remainder do they leave when divided by another number? You already know this idea informally — classifying integers as even or odd is really just checking whether their remainder on division by $2$ is $0$ or $1$. The same instinct scales up: knowing only the remainder a number leaves, rather than its full value, is often all you need to solve a problem. Consider a simple puzzle — if today is a Wednesday, what day of the week falls $68$ days from now? Since a week repeats every $7$ days, only the remainder of $68 \div 7$ matters: dividing $68$ by $7$ leaves a remainder of $5$, so the answer is $5$ days after Wednesday, which is Monday. This chapter formalises this remainder-based way of thinking about numbers — starting with modular arithmetic, the system of arithmetic built entirely around remainders, before moving on to numerical applications used in everyday business and financial situations.

Illustration 1. If today is Wednesday, what day will it be after 68 days?

A week is cyclic — the seven days repeat after every 7 days — so we only need the remainder when 68 is divided by 7:

$68 = 7 \times 9 + 5 \quad\Rightarrow\quad 68 \bmod 7 = 5.$ …