Congruence by Product of Moduli/Real Modulus

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

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

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

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


Let $y \in \R: y \ne 0$.


\(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod z\)
\(\ds \leadstoandfrom \ \ \) \(\ds a \bmod z\) \(=\) \(\ds b \bmod z\) Definition of Congruence
\(\ds \leadstoandfrom \ \ \) \(\ds y \paren {a \bmod z}\) \(=\) \(\ds y \paren {b \bmod z}\) Left hand implication valid only when $y \ne 0$
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {y a} \bmod \paren {y z}\) \(=\) \(\ds \paren {y b} \bmod \paren {y z}\) Product Distributes over Modulo Operation
\(\ds y a\) \(\equiv\) \(\ds y b\) \(\ds \pmod {y z}\) Definition of Congruence

Hence the result.

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