Congruence by Product of Moduli

Theorem

Let $a, b, m \in \Z$.

Let $a \equiv b \pmod m$ denote that $a$ is congruent to $b$ modulo $m$.

Then $\forall n \in \Z, n \ne 0$:

$a \equiv b \pmod m \iff a n \equiv b n \pmod {m n}$

Real Modulus

Let $a, b, z \in \R$.

Let $a \equiv b \pmod z$ denote that $a$ is congruent to $b$ modulo $z$.

Then $\forall y \in \R, y \ne 0$:

$a \equiv b \pmod z \iff y a \equiv y b \pmod {y z}$

Proof

Let $n \in \Z: n \ne 0$.

Then:

 $\displaystyle a$ $\equiv$ $\displaystyle b$ $\displaystyle \pmod m$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a \bmod m$ $=$ $\displaystyle b \bmod m$ Definition of Congruence Modulo Integer $\displaystyle \leadstoandfrom \ \$ $\displaystyle n \left({a \bmod n}\right)$ $=$ $\displaystyle n \left({b \bmod n}\right)$ Left hand implication valid only when $n \ne 0$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \left({a n}\right) \bmod \left({m n}\right)$ $=$ $\displaystyle \left({b n}\right) \bmod \left({m n}\right)$ Product Distributes over Modulo Operation $\displaystyle \leadstoandfrom \ \$ $\displaystyle a n$ $\equiv$ $\displaystyle b n$ $\displaystyle \pmod {m n}$ Definition of Congruence Modulo Integer

Hence the result.

Note the invalidity of the third step when $n = 0$.

$\blacksquare$