# Quadratic Residues

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Fix a positive integer a We say that a is a quadratic residue modulo n if there exists x such that a = x^2 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{O.1....,n?1} satisfying x^2=amodn.

(b) Show that if a 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, n such that....

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Proof:

a. Suppose and (mod ), then we have (mod ). This implies that . But is an odd prime. Then we have or . This ...

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Quadratic residues are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.