Consider the statement: For all integers $r$, $s$, and $a$, and natural numbers $m$, if $ra \equiv sa \pmod m$ then $r\equiv s \pmod m$.
I have found this statement to be false by the counterexample where $s$ is a negative number or $a$ is equal to zero.
the salvage wouldn't be a problem if I could change the quantifiers but the question specifically says:
If it is false, provide a counterexample and attempt to salvage the result. In other words, without changing the quantifiers (maintaining a universal statement) determine a different conclusion that makes the statement true or suggest additional hypotheses so that the conclusion follows.
I have messed around and tried to make the statement read if $ra \equiv sa \pmod m$ and $sa>0$ then $r\equiv s \pmod m$.
But I have found that it still fails, and at this point im really stumped and don't know of any conclustion or additional hypotheses I can make of that statement that would cause it to be true.