Problem: Let p and q be distinct primes. What is the maximum number of possible solutions to a congruence of the form $x^2-a \equiv 0$ (mod pq), where as usual we are only interested in solutions that are distinct modulo pq? (Also provide a proof to support your conjecture.)
Attempt 1: I believe that the answer is p when $p>q$ or q when $q>p$. Am I right or wrong? If I am right can you please help me proof the statement?
I was wrong.
Attempt 2: There are at most 2 solutions because of the Polynomial Roots Mod p Theorem.