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M135F07A6 - MATH 135 Assignment#6 Hand-In Problems 1 In...

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MATH 135 Fall 2007 Assignment #6 Due: Wednesday 31 October 2007, 8:20 a.m. Hand-In Problems 1. In each part, explain how you got your answer. (a) What is the remainder when 14 585 is divided by 3? (b) Is 8 24 + 13 12 divisible by 7? (c) What is the last digit in the base 6 representation of 8 24 ? (d) Determine the remainder when 42 2007 + 2007 10 is divided by 17. 2. Suppose that p is a prime number. (a) Prove that if x y (mod p ), then x n y n (mod p ) for every n P by induction on n . (This proof is not difficult mathematically. We will be looking for a very carefully written proof here.) (b) Using the definition of congruence, prove that if x 2 y 2 (mod p ), then x ≡ ± y (mod p ). (c) Determine the number of integers a with 0 a < 4013 with the property that there exists an integer m with m 2 a (mod 4013). (Note that 4013 is a prime number.) (d) Disprove the statement “If x 4 y 4 (mod p ), then x ≡ ± y (mod p )”. 3. (a) Prove that 5 n 7 + 7 n 5 + 23 n 0 (mod 7) for all n P .
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