$xa \equiv ya \pmod p $ implies that $x \equiv y \pmod p$

Question

We are working in the set of integers.

For $x,y,p,a$: $$x \equiv y \pmod p \implies xa \equiv ya \pmod p$$

When can we also assume the following? $$ xa \equiv ya \pmod p \implies x \equiv y \pmod p $$

Another way to ask the question is to prove that for $a,b$ numbers $a,2a,3a,...,ba$ are all different $\pmod b$. I am kind of confused since $x^2$ can return same remainder $\pmod a$ for different $x \pmod a$

Could be, not necesarily. I am asking generally for what numbers can we assume or can we divide by $a$. — Csaba Daniel Farkaš, Jun 14 '20 at 09:05
In every integral domain $\mathbf A$ it hold the cancellation proprierty i.e. $$ac = bc \implies a = b $$for every $a,b \in \mathbf A$ and $c \neq 0_{\mathbf A}$. When is $\frac{\mathbb Z}{p\mathbb Z}$ an integral domain? — Sewer Keeper, Jun 14 '20 at 09:07

score 2 · Answer 1 · edited Jun 14 '20 at 09:11

2

Let $gcd(a,p)=1$, then: $$\begin{align} xa\equiv ya \pmod p &\implies (x-y)a \equiv 0 \pmod p\\ & \implies x\equiv y \pmod p\end{align}$$

edited Jun 14 '20 at 09:11

Csaba Daniel Farkaš

543

answered Jun 14 '20 at 09:08

Bcpicao

908

Thank you, with this method I can finish the rest! – Csaba Daniel Farkaš Jun 14 '20 at 09:09

$xa \equiv ya \pmod p $ implies that $x \equiv y \pmod p$

1 Answers1