Divisibility

An integer $a$ divides an integer $b$ (written $a \mid b$) if there exists an integer $k$ such that $b = ak$. When $a \mid b$, we say $a$ is a divisor or factor of $b$, and $b$ is a multiple of $a$.

Divisibility defines a partial order on the positive integers: it is reflexive ($a \mid a$), antisymmetric (if $a \mid b$ and $b \mid a$ then $a = b$ for positive integers), and transitive (if $a \mid b$ and $b \mid c$ then $a \mid c$). Under this order, the positive integers form a lattice where meet is the greatest common divisor and join is the least common multiple.