H-02 - statement that if a ¬ n x = b has a unique solution...

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Second Homework Assignment Write the solution to each question on a single page. The deadline for handing in solutions is 22 February 2010. Question 1. (20 = 10 + 10 points). (Problem 2.1-12 in our textbook). We recall that a prime number p , that divides a product of integers, divides one of the two fac- tors. (a) Let 1 a p 1. Use the above recol- lection to show that as b runs through the integers from 0 to p 1, the products a · p b are all di±erent. (b) Explain why every positive integer less than p has a unique multiplicative inverse in Z p . Question 2. (20 points). (Problem 2.2-22 in our textbook). Either ²nd an equation of the form a · n x = b in Z n that has a unique solution even though a and n are not relatively prime, or prove that no such equation exists. In other words, either prove the
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Unformatted text preview: statement that if a ¬∑ n x = b has a unique solution in Z n , then a and n are relatively prime, or ¬≤nd a counterexample. Question 3. (20 = 10 + 10 points). (Problem 2.2-17 in our textbook). Recall the Fibonacci numbers de¬≤ned by F = 0, F 1 = 1, and F i = F i-1 + F i-2 for all i ‚Č• 2. (a) Run the extended gcd algorithm for j = F 10 and k = F 11 , showing the values of all parameters at all levels of the recursion. (b) Running the extended gcd algorithm for j = F i and k = F i +1 , how many recursive calls does it take to get the result? Question 4. (20 points). Let n ‚Č• 1 be a nonprime and x ‚ąą Z n such that gcd( x, n ) n = 1. Prove that x n-1 mod n n = 1 . 1...
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