Modular Arithmetic

Notation	Meaning
$m \mid n$	$m$ divides $n$
$m \nmid n$	$m$ does not divide $n$
$n \equiv r (mod m)$	$m \mid (n-r)$, that is, as $m$ divides $n-r$, $n$ and $r$ have the same remainder when divided by $m$
$\lfloor x \rfloor$	floor function: largest integer not larger than x
$n mod m$	(binary operator) remainder of $n$ when divided by $m$ ($m$ is called the modulus, which we assume here to be a positive integer). Equal to $n-m \lfloor \frac{n}{m} \rfloor$. Note that $0 \le r \lt m$. In C, Python, Java, and pari-gp, it can be computed using using the % binary operator (applied to unsigned integers).
$gcd(a,b)$	gratest common divisor of $a$ and $b$.
$lcm(a,b)$	least common multiple of $a$ and $b$; equal to $ab/ gcd(a,b)$.
$\mathbb{Z}_m$	set of equivalence classes modulo $m$; slightly abusing the mathematical notation for equivalence classes, $\mathbb{Z}_m = \lbrace 0, 1, ..., m-1 \rbrace$.

Examples