math 311w homework3

math 311w homework3 - linear combination nx + my also...

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Homework 3. Due Friday, February 5, 2010 Give rigorous solutions of all problems. Problem 1. Can number 15 be written as 111 n + 180 m ? If it can, then show how. Problem 2. We denote as F n the n -th Fibonacci’s number, where F 1 = 1 ,F 2 = 1 ,F n +1 = F n + F n - 1 . Prove by induction that every two successive terms of Fibonacci’s numbers are relatively prime. Problem 3. Assume that the set S is a subset of integer numbers. Assume it has a property that if numbers x and y belong to S then their integer
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Unformatted text preview: linear combination nx + my also belongs to S . Prove, that a) Either S consists from one number zero or there is a positive number that belongs to S . b) Either S consists from one number zero or there is a positive number d such that any other number in S is proportional to d . Problem 4. Find the greatest common divisor of a and b , and represent it in the form d = na + mb . a) a = 91 ,b = 126 b) a = 630 ,b = 132 1...
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