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Unformatted text preview: MATH 135 Fall 2008 Assignment #6 Due: Wednesday 29 October 2008, 8:20 a.m. HandIn Problems 1. In parts (a) to (d), state the answer. No justification is necessary. In parts (e) to (h), explain how you got your answer. (a) Is 14 ≡ 22 (mod 18)? (b) Is 2473 ≡  26 (mod 17)? (c) Is 8 16 ≡ 4 (mod 12)? (d) For how many n ∈ P with 1 ≤ n ≤ 2008 is n ≡ 0 (mod 4)? (e) What is the remainder when 8 243 is divided by 3? (f) Is 8 24 + 13 12 divisible by 7? (g) Determine the remainder when 3 47 5 74 + ( 9) 10 is divided by 13. (h) For how many m ∈ P with m > 1 is 14 ≡  58 (mod m )? 2. Suppose that a,b ∈ Z and m ∈ P , with a ≡ b (mod m ). Prove by induction that a n ≡ b n (mod m ) for every n ∈ P . 3. Suppose that a,b ∈ Z and m,n ∈ P . Prove that an ≡ bn (mod mn ) if and only if a ≡ b (mod m ). 4. Suppose that p is a prime number with p > 3. (a) By consider the possible remainders when p is divided by 4, prove that p ≡ 1 or 3 (mod 4)....
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This note was uploaded on 09/04/2009 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.
 Fall '08
 ANDREWCHILDS
 Math, Algebra

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