If x = 0 (mod p) then the only solution in Zp for x will be x = 0, since this is the only x in Zp that is divisible by p.

Similarly, if x = -1 (mod p) then the only solution in Zp for x will be x = p - 1, because this is the only x in Zp that will leave a residue of -1.

-Dan