1. The problem statement, all variables and given/known data If a and b are both quadratic residues/nonresidues mod p & q where p and q are distinct odd primes and a and b are not divisible by p or q, Then x2 = ab (mod pq) 2. Relevant equations Legendre symbols: (a/p) = (b/p) and (a/q) = (b/q) quadratic residue means x2 = a (mod p) 3. The attempt at a solution I know that since (a/p) = (b/p) and (a/q) = (b/q) that (ab/p) = 1 and (ab/q) = 1 (ab is a quadratic residue for both mod p and mod q). So I have x12=ab(mod p) and x22=ab(mod q). The Chinese remainder theorem takes me in circles, and since I can't guarantee x12= x22, then I can't just multiply p and q and call it good. I've spent hours trying to find a counter example since I can't figure out how to combine p and q into the same mod. So I'm stuck. any hints on what to look at for either a proof or counterexample?