2007finalsol

2007finalsol - Math 465 Final(150 pts May 9 2007 Print Name...

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Unformatted text preview: Math 465, Final (150 pts) May 9, 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 110 minutes to finish this exam. 1. SHORT ANSWER ( no explanation required .) (a) (5pts) Evaluate d (585) , σ (585) , φ (585) , μ (585). Solution. 585 = 3 2 · 5 · 13 d (585) = (2 + 1)(1 + 1)(1 + 1) = 12 σ (585) = 3 3- 1 3- 1 5 2- 1 5- 1 13 2- 1 13- 1 = 1092 φ (585) = φ (3 2 ) φ (5) φ (13) = (3 2- 3)(5- 1)(13- 1) = 288 μ (585) = 0 (b) (5 pts) How many primitive roots are there modulo 36? Solution. 36 = 2 2 · 3 2 So, there are no primitive roots. (c) (5 pts) Can 84500 be expressed as a sum of two squares? (Note that 84500 = 2 2 · 5 3 · 19 2 .) Solution. Since 19 ≡ 3 (mod 4) and the power of 19 is 2, 84500 can be expressed as a sum of two squares. (d) (5 pts) Determine whether or not 8911 is a Carmichael number. (Note that 8911 = 7 · 19 · 67 and 8910 = 2 · 3 4 · 5 · 11.) Solution. 8911 is a Carmichael number, since 8911 = 1 · 19 · 67 and 8910 is divisible...
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2007finalsol - Math 465 Final(150 pts May 9 2007 Print Name...

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