Can we say If q is incongruent to p modulo n then $q\equiv -p$ (mod n)

Question

Am I right to write:

If q is incongruent to p modulo n, then $q\equiv -p$ (mod n)

Thanks for helping

Answer 1

No. The relation $q\equiv -p \pmod{n}$ means that $p + q\equiv 0\pmod{n}$. To say that $q$ is inequivalent to $p$ means that $q\not\equiv p\pmod{n}$; here, $q$ could be any of the $n-1$ values aside from $p\pmod{n}$.

Answer 2

Consider this:

3 is not cong. to -1 mode 6 and 3 is not cong. to +1 mode 6.