The following proof needs verification:
Let $a, b, r \in [0, m-1]$ be integers, $m$ a prime, where
$$ r = ab - qm $$
is a non-negative remainder, and $q$ some non-negative integer.
This is where the proof starts:
Given $a$, assume there are two distinct $b \neq b'$ and $q \neq q'$ such that
$$ ab - qm = ab' - q'm. $$
Then
$$ a(b - b')/(q - q') = m $$
is either wrong for $b - b' \le q - q'$, or $m$ is not a prime, since a factorization exists.
Hence, if $m$ is prime, for all two $b \neq b'$ the remainders are different.