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MAT135-Assignment8-GradingScheme

# MAT135-Assignment8-GradingScheme - MATH 135 Fall 2010...

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MATH 135, Fall 2010 Solution of Assignment #8 Problem 1 . (a) Prove that 2 n 25 + 10 n 13 + 14 n 9 + 9 n 0 (mod 5) for all n Z . (b) Prove that 35 | 2 n 25 + 10 n 13 + 14 n 9 + 9 n for all n Z Solution. (a) By Fermat’s Little Theorem, n 5 n (mod 5) for all n P . Thus, 2 n 25 + 10 n 13 + 14 n 9 + 9 n 2( n 5 ) 5 + 14 n 5 · n 4 + 9 n (mod 5) 2 n 5 + 14 n 5 + 9 n (mod 5) 25 n (mod 5) 0 (mod 5) as required. (b) Using (a), we already know that 5 | 2 n 25 + 10 n 13 + 14 n 9 + 9 n for all n Z . To prove that 35 | 2 n 25 + 10 n 13 + 14 n 9 + 9 n for all n P , we must prove that 2 n 25 + 10 n 13 + 14 n 9 + 9 n 0 (mod 7) for all n P . By Fermat’s Little Theorem, n 7 n (mod 7) for all n P . Thus, 2 n 25 + 10 n 13 + 14 n 9 + 9 n 2 n 7 · n 7 · n 7 · n 4 + 10 n 7 · n 6 + 9 n (mod 7) 2 n 7 + 10 n 7 + 9 n 21 n (mod 7) 0 (mod 7) Thus, 2 n 25 + 10 n 13 + 14 n 9 + 9 n 0 (mod 7) for all n P , so 35 | 2 n 25 + 10 n 13 + 14 n 9 + 9 n for all n P . Problem 2 . Chinese generals used to count their troops by telling them to form groups of some size n , and then counting the number of troops left over. Suppose there were 10000 troops before

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