Modulo Multiplication on Reduced Residue System is Cancellable

Theorem
Let $m \in \Z_{> 0}$ be a (strictly) positive integer.

Let $\Z'_m$ be the reduced residue system modulo $m$:


 * $\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$

Let $S = \struct {\Z'_m, \times_m}$ be the algebraic structure consisting of $\Z'_m$ under modulo multiplication.

Then $\times_m$ is cancellable, in the sense that:
 * $\forall a, b, c \in \Z'_m: a \times_m c = b \times_m c \implies a = b$

and:
 * $\forall a, b, c \in \Z'_m: c \times_m a = c \times_m b \implies a = b$

Proof
Let $a, b, c \in \Z'_m$ such that $a \times_m c = b \times_m c$

Let $p, q, r$ be integers such that:
 * $p \in a$
 * $q \in b$
 * $r \in c$

By definition of residue class, this means:

Thus as $p \in a$ and $q \in b$ it follows that $a = b$.

Hence the result.