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Unformatted text preview: Let p be a prime, a be an integer with ( a,p ) = 1. a) (2 pts) Show that { a, 2 a, , ( p1) a } is a reduced residue system for the modulus p . b) (2 pts) Let N k = 1 k + 2 k + + ( p1) k . Use the results of part a) to show that a k N k N k mod p 1 2 MATH 115 MIDTERM 1 for any a with ( a,p ) = 1. Note that the result of part b) can be written as ( a k1) N k mod p for any ( a,p ) = 1. c) (1 pt) In part b), take the numbner a to be a primitive root g mod p . Show that N k mod p whenever k is not divisible by p1. (hint: since g is a primitive root modulo p , we have ord p ( g ) = p1.)...
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 Fall '07
 MOK
 Math, Number Theory

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