Congruence of Quotient/Proof 2

Theorem

Let $a, b \in \Z$ and $n \in \N$.

Let $a$ be congruent to $b$ modulo $n$, that is $a \equiv b \pmod n$.

Let $d \in \Z: d > 0$ such that $d$ is a common divisor of $a, b$ and $n$.

Then:

$\dfrac a d \equiv \dfrac b d \pmod {n / d}$

Proof

From Congruence by Product of Moduli, we have that:

$\dfrac a d \equiv \dfrac b d \pmod {n / d} \iff d \dfrac a d \equiv d \dfrac b d \pmod {n / d} \iff a \equiv b \pmod n$

$\blacksquare$