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Unformatted text preview: Math 465, Exam II (100 pts) April 7, 2008 Print Name: Ae Ja Yee This exam is closedbook, closednotes. Show all your work for full credit unless other wise indicated. Partial credit will be given based on what is written. You have 50 minutes to finish the exam. 1. SHORT ANSWER (No explanation required.) (a) (10 pts) How many quadratic residues are there modulo 101? (Note that 101 is a prime number.) Solution. 50. (b) (10 pts) Are there primitive roots modulo 36? Solution. No. (c) (10 pts) How many primitive roots are there modulo 81? Solution. φ ( φ (3 4 )) = φ (54) = φ (2) φ (27) = 27 9 = 18. 2. Evaluate the following Legendre symbols. (a) (10 pts) ( 5 19 ) (Use Gauss’ lemma.) (b) (10 pts) ( 1234 4567 ) ( Use the quadratic reciprocity law.) Solution. (a) Consider 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45. Then 5 , 10 , 15 , 20 ≡ 1 (mod 19) , 25 ≡ 6 (mod 19) , 30 ≡ 11 (mod 19) , 35 ≡ 16 (mod 19) , 40 ≡ 2 (mod 19) , 45 ≡ 7 (mod 19) ....
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 Spring '08
 YEE
 Math, Number Theory, Prime number, 3k, 2k, 10 pts, 50 Minutes

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