finalsol - Math 465, Final (150 pts) Print Name: May 5,...

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Math 465, Final (150 pts) May 5, 2008 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 110 minutes to finish this exam. 1. SHORT ANSWER ( No explanation required .) (a) (10 pts) Evaluate the Legendre symbol ( - 8 31 ) . (Note that 31 is a prime number.) Solution. ± - 8 31 ² = ± - 1 31 ²± 2 31 ² 3 = ± - 1 31 ²± 2 31 ² = - 1 · 1 = - 1 (b) (10 pts) How many primitive roots are there modulo 27? Solution. Since 27 = 3 3 , there are φ ( φ (3 3 )) primitive roots, namely φ (3 2 (3 - 1)) = (3 2 - 3) = 6. (c) (10 pts) Can 3500 be expressed as the sum of two squares? Solution. Note that 3500 = 10 2 · 5 · 7. Since 7 3 (mod 4), 3500 cannot be expressed as the sum of two squares. (d) (10 pts) Can 3500 be expressed as the sum of three squares? Solution. Since 3500 = 4(8 · 109 + 3), 3500 can be expressed as the sum of three squares. 2. (10 pts) Use the Euclidean algorithm to compute (195 , 25) and express (195 , 25) in the form 195 x + 25 y . Solution.
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finalsol - Math 465, Final (150 pts) Print Name: May 5,...

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