Modular Arithmetic

Modular arithmetic deals with remainders. The expression $a \equiv r \pmod{b}$ means that when $a$ is divided by $b$, the remainder is $r$. A key property is that you can perform algebra on the remainders themselves.

This is the basis for divisibility rules. For example, since $10 \equiv 1 \pmod{9}$, any power of 10 is also congruent to 1. Therefore, a number $n = d_k 10^k + \dots + d_1 10^1 + d_0 10^0$ is congruent to the sum of its digits $(d_k + \dots + d_0) \pmod{9}$.

To solve problems like 'Find $x$ where $x+3 \equiv 9 \pmod{5}$', you can first simplify the known numbers: $9 \equiv 4 \pmod{5}$. The problem becomes $x+3 \equiv 4 \pmod{5}$, which simplifies to $x \equiv 1 \pmod{5}$.