I want to show that , in a given pair, there exists a pair where $x-y \equiv 0 \pmod n$ and in order to do that I can write an algorithm that looks for two different instances where we have the same remainder in $\pmod n$. So that we would have $0 \pmod n$, correct?
So can I then just turn this question into this: Prove that $a \equiv b \pmod n$ if and only if $a$ and $b$ leave the same remainder when divided by $n$? And how can I prove this?
