Quadratic residues. Fix a positive integer N. We say that a is a quadratic residue modulo N if there exists x such that a = x2 mod N.
(a) Let N be an odd prime and a be a non-zero quadratic residue modulo N. Show that there are exactly two values in {0,1,..., N − 1} satisfying x2 ≡ a mod N.
(b) Show that if N is an odd prime, there are exactly (N + 1)/2 quadratic residues in {0,1,..., N − 1}.
(c) Give an example of positive integers a and N such that x2 = a mod N has more than two solutions in {0,1,...,N − 1}.
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