Modulo Multiplication is Closed

Theorem

Multiplication modulo $m$ is closed on the set of integers modulo $m$:

$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m \in \Z_m$.

Proof

From the definition of multiplication modulo $m$, we have:

$\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$

By the Division Theorem:

$x y = q m + r$ where $0 \le r < m$

Therefore for all $0 \le r < m$:

$\eqclass {x y} m = \eqclass r m$

Therefore from the definition of integers modulo $m$:

$\eqclass {x y} m \in \Z_m$

$\blacksquare$