Given a rational number $\frac{a}{b}$ and a prime $p$ such that $p\not\mid b$, we say that $\frac{a}{b} \equiv 0 \pmod p$ if $p \mid a$. Also, we say for $\alpha,\beta \in \mathbb{Q}$, $\alpha \equiv \beta \pmod p$ if $\alpha - \beta \equiv 0$.
Now, I have to questions:
What should I think if I consider modulo $p$ equivalence of some rational number $\alpha$? Is it a number between $0$ and $p-1$ such that it is $0$ if above holds and some number different than $0$ otherwise?
How can I show the following:
Given a rational $\alpha \equiv 0 \pmod p$, there exists a unique number $q$, $0 \le q < p$ such that $q \equiv \frac{1}{p} \alpha \pmod p$?
Hint.- Simply $\dfrac ab=\dfrac {ab^{-1}}{bb^{-1}}=ab^{-1}$ and $\dfrac ab\equiv0\iff \dfrac ab=Mp$