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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- Spring '10
- Prime number, Euclidean algorithm, unique multiplicative inverse, extended gcd algorithm