Prove each of the following. (A) Prove that if a and b are quadratic residues modulo m, then ab is also a quadratic residue mod m. (B) Prove that the square of a quadratic nonresidue of an odd prime p is a quadratic residue. (C) Prove that the product of a quadratic residue and a quadratic nonresidue of an odd prime p is a quadratic nonresidue.
Prove each of the following. (A) Prove that if a and b are quadratic residues modulo m, then ab is also a quadratic residue mod m. (B) Prove that the square of a quadratic nonresidue of an odd prime p is a quadratic residue. (C) Prove that the product of a quadratic residue and a quadratic nonresidue of an odd prime p is a quadratic nonresidue.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:10 Prove each of the following.
(A) Prove that if a and b are quadratic residues modulo m, then ab is also a quadratic residue mod m.
(B) Prove that the square of a quadratic nonresidue of an odd prime p is a quadratic residue.
(C) Prove that the product of a quadratic residue and a quadratic nonresidue of an odd prime p is a
quadratic nonresidue.
(D) If a has order hk (mod m), then a" has order k (mod m).
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