Cancellability of Congruences/Corollary 2

Corollary to Cancellability of Congruences

Let $a, b, c$ be integers.

Let $p$ be a prime number such that $p \nmid c$.

Then:

$c a \equiv c b \pmod p \implies a \equiv b \pmod p$

where $\equiv$ denotes congruence.

Proof

As $p \nmid c$, it follows from Prime not Divisor implies Coprime that:

$p \perp c$

where $\perp$ denotes that $p$ and $c$ are coprime.

The result follows from Cancellability of Congruences: Corollary 1.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $4$