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math 311w homework4

math 311w homework4 - p is prime and it divides a product...

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Homework 4. Due Friday, February 12, 2009 Give rigorous solutions of all problems. Problem 1. Find a solution for the following equations if it exists a) 357 x + 182 y = 36 b) 123 x + 347 y = 15 Problem 2. Prove that there are infinitely many prime numbers. Problem 3. Prove that every integer number n 2 could be written as a product of prime numbers. Problem 4. Prove by induction that if a number
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Unformatted text preview: p is prime and it divides a product of n numbers a 1 · a 2 · ... · a n , then it divides one of the numbers a i . You can assume that it is true when there are only two numbers. Problem 5. a) Prove that for odd n a n + b n = ( a + b )( a n-1-a n-2 b + a n-3 b 2- ··· + (-1) n-1 b n-1 ) b) Prove that if 2 n + 1 is prime, then n = 2 k . 1...
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