Math 115 - Fall 2000 - Ribet - Midterm 1

Math 115 - Fall 2000 - Ribet - Midterm 1 - 2 . Find the...

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Math 115 Professor K. A. Ribet First Midterm Exam September 29, 2000 This is a closed-book exam: no notes, books or calculators are allowed. Explain your answers in complete English sentences. No credit will be given for a “correct answer” that is not explained fully. In general, there is no need to simplify numerical answers. 1 (5 points) . Let a and b be positive integers for which a 4 divides b 3 . Prove that a divides b . 2 (10 points) . Let f ( x ) = x 2 - x - 1. Here are some values of f : i 0 1 2 3 4 5 6 7 8 9 10 ··· f ( i ) - 1 - 1 1 5 11 19 29 41 55 71 89 ··· . Find integers a and b so that f ( a ) and f ( b ) are both divisible by 11 2 but so that a - b is not divisible by 11
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Unformatted text preview: 2 . Find the number of solutions mod 5 11 2 to the congruence f ( x ) 0 mod 5 11 2 . 3 (3 points) . Let m = 173 193. Find positive integers a and b with m < b < m + 1 2 for which m = b 2-a 2 . 4 (5 points) . Use the identity 1 = 89 24-61 35 ( * ) to solve the simultaneous congruences x n 3 mod 89 12 mod 61. 5 (4 points) . Using (*), nd integers a and b with 1 = 24 a +35 b and | a | as small as possible. 6 (3 points) . Using (*) yet again, solve the congruence 35 x 2 mod 89....
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