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Unformatted text preview: Math 465, Final (150 pts) May 9, 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 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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 Spring '08
 YEE
 Math, Number Theory, pts, Carmichael

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