elementary number theory - Congruence proof using gcd concepts

Anyone can help me? The problem is:

We have that $a \equiv b (\mod p)$, $x|a, x|b$, and $x$ and $p$ are relative primes, $\gcd(x,p)=1$. How to show that $\dfrac{a}{x} \equiv\dfrac{b}{x} (\mod p)$?

Thanks in advance.

Answer

Hint $\ $ By Euclid's Lemma, $\rm\, gcd(x,p)=1,\ \ x\mid \color{#c00}{a\!-\!b = pn}\:\Rightarrow\:x\mid n.\ $ Cancel $\rm\,x\,$ from $\rm\color{#c00}{a,b,n\ there}$.