Modular congruence ($\equiv$)

The congruence $a\equiv b(\mathrm{mod}n)$ means that $n$ divides $a-b$.

For a prime $p$ and $k\ge 1$, write $a=b+p^{k}h$. The binomial theorem shows $a^p-b^p$ is divisible by $p^{k+1}$: the linear term contains the extra factor $p$ and every higher term contains at least $p^{2k}$. Iteration gives the displayed modular congruence for every $r\ge 0$, including $p=2$. Applied to Fermat's little theorem, it yields $a^{p^n}\equiv a^{p^{n-1}}(\mathrm{mod}{p}^n)$.

For a positive modulus $n$, a reduced residue system contains exactly one representative of each residue class coprime to $n$. Its size is the Euler totient function $\phi (n)$. Multiplication by any unit modulo n permutes these classes, which proves the Fermat-Euler theorem after multiplying and cancelling their unit product.

A linear congruence has the form $ax\equiv b(\mathrm{mod}n)$. It has a solution exactly when $\gcd (a,n)$ divides $b$.

The residue class $[a{]}_m$ consists of all integers congruent to $a$ modulo $m$.

A residue class modulo $n$ is a unit exactly when it is represented by an integer coprime to $n$. The units form the multiplicative group $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.