Definition:Modulo Multiplication/Definition 3

Definition
Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \set {0, 1, \ldots, m - 1}$

The operation of multiplication modulo $m$ is defined on $\Z_m$ as:
 * $a \times_m b := a b - k m$

where $k$ is the largest integer such that $k m \le a b$.

Also see

 * Equivalence of Definitions of Modulo Multiplication