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2007exam2sol

# 2007exam2sol - MATH 465 Exam II(100 pts Print Name April 6...

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MATH 465, Exam II (100 pts) April 6, 2007 Print Name: Ae Ja Yee This exam is closed-book, closed-notes. 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) (5 pts) Evaluate the Legendre symbol ( - 8 101 ) . Solution. ( - 8 101 ) = ( - 1 101 ) ( 2 101 ) 3 = - 1 (b) (5 pts) How many quadratic residues are there modulo 307? (Note that 307 is a prime number.) Solution. (307 - 1) / 2 = 153. (c) (5 pts) Does 28 have primitive roots? Solution. No. Only 1 , 2 , 4 , p k , and 2 p k for p prime have primitive roots. (d) (5 pts) How many primitive roots are there modulo 50? Solution. φ ( φ (50)) = φ (20) = 8. 2. Evaluate the following Legendre symbols. (a) (10 pts) ( 3 13 ) (Use Euler’s criterion.) (b) (10 pts) ( 6 11 ) (Use Gauss’ lemma.) (c) (10 pts) ( 41 83 ) (Note that 41 and 83 are prime numbers. Use Gauss’ reciprocity law.) Solution. (a) By Euler’s criterion, 3 13 3 (13 - 1) / 2 = (3 3 ) 2 1 (mod 13) . (b) Consider 6 , 12 , 18 , 24 , 30. Then 6 , 12 1 (mod 11) , 18 7 (mod 11) , 24 2 (mod 11) , 30 8 (mod 11) .

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