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Unformatted text preview: MATH 465, Exam II (100 pts) April 6, 2007 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) (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,...
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This note was uploaded on 07/23/2008 for the course MATH 465 taught by Professor Yee during the Spring '08 term at Penn State.
 Spring '08
 YEE
 Math, Number Theory

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