Congruence by Product of Moduli

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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 $n = 0$.

$\blacksquare$


Sources