Determine whether or not the following congruence has a solution: x^2 = 78 (mod 41503). If a solution exist then find a positive integer that satisfies the solution. Here, = means "congruent to". I understand the fact that if 78 is the quadratic residue of 41503 then we have a solution but I don't know how I should do that? Should I break down 78 into factors of prime and then try to verify using quadratic reciprocity? Assuming that there IS a solution, how do I find an integer that satisfies the congruent? Please help! thanks!