HW10-solutions - EECS 203: DISCRETE MATHEMATICS Homework 10...

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EECS 203: DISCRETE MATHEMATICS Homework 10 Solutions 1. (4 points) Section 3.5, Problem 8. Consider the n integers starting with ( n +1)!+2 , ( n +1)!+3 , . . . , ( n +1)!+( n +1) where n Z + . Notice that k | ( n +1)! for 1 k n +1. Thus, 2 | [( n +1)!+2], 3 | [( n +1)!+3] , . . . , ( n +1) | [( n +1)!+( n +1)] and each of these numbers is composite. 2. (4 points) Calculate 3 530 mod 53. Indicate which theorems or properties of modular arithmetic you are using to arrive at an answer. Since 53 is prime we may apply Fermat’s little theorem. 3 530 (3 52 ) 10 · 3 10 (mod 53) [(1) 10 mod 53] · [3 10 mod 53] (mod 53) [(3 2 ) 5 mod 53] (mod 53) [(9) 2 mod 53][(9) 3 mod 53] (mod 53) [28][40] (mod 53) [56][20] (mod 53) [3][20] (mod 53) 7 (mod 53) 3. (4 points) Prove that 2 a + 1 and 4 a 2 + 1 are relatively prime. Let’s assume a N so that 2 a + 1 and 4 a 2 + 1 are non-negative. We use the Euclidean algorithm to find gcd(2 a + 1 , 4 a 2 + 1). Notice that (2
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This note was uploaded on 12/20/2010 for the course EECS 203 taught by Professor Yaoyunshi during the Fall '07 term at University of Michigan.

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HW10-solutions - EECS 203: DISCRETE MATHEMATICS Homework 10...

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