Youaresuggestedtodoallofthefollowingproblems. M210-050 D

Youaresuggestedtodoallofthefollowingproblems.
M210-050 Discrete Mathematics Homework 3/4 You are suggested to do all of the following problems. Please hand in only three of them for grading. 1. Determine d :=gcd(2009; 42) by using Euclidean algrithm and nd integers u; v such that d = 2009u + 42v. 2. Let a; b; c be integers such that ajc bjc and gcd(a; b) = 1. Show that abjc. 3. Suppose a x are integers such that a > 1 aj(4x+7) aj(16x+5). Find a and determine x (mod a) (i.e. the remainder when x is divided by a). 4. For any k 2 N show that gcd(3k + 2; 6k2 + 13k + 3) = 1. 5. Suppose that p p + 4 are primes and p > 3. Show that 6j(p ?? 1). 6. If n is an integer coprime with 3 show that x2 + y2 = 3n has no integer solutions. 1
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