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Fsum11_solutions - MAT 312/AMS 351 Solutions to Final Exam...

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MAT 312/AMS 351 Solutions to Final Exam. 1. (30 points) a) (10 points) Solve the system of congruences x = 10 mod 13 , x = 16 mod 19 . Answer: x = 244 mod 247. Solutions: 1) Remark that x = - 3 mod 13 , x = - 3 mod 19 , so x = - 3 mod 13 · 19 , x = 244 mod 247 . 2) Since 13 and 19 are coprime, we can present 1 as their linear combi- nation using Euclidean algorithm: 19 = 13 + 6 , 13 = 2 · 6 + 1 , 6 = 19 - 13 , 1 = 13 - 2(19 - 13) = 3 · 13 - 2 · 19 . Therefore x = 3 · 13 · 16 - 2 · 19 · 10 = 624 - 380 = 244 . b) (10 points) Solve the equation 14 x = 6 mod 21 . Answer: No solutions. Solution: We have d = g.c.d (14 , 21) = 7 and 6 is not divisible by d . Therefore the equation has no solutions. c) (10 points) Solve the equation 14 x = 6 mod 31 . . Answer: x = 27 mod 31 . 1
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Solution: Since 14 and 31 are coprime, we can present 1 as their linear combination using Euclidean algorithm: 31 = 2 · 14 + 3 , 14 = 4 · 3 + 2 , 3 = 2 + 1 , 3 = 31 - 2 · 14 , 2 = 14 - 4(31 - 2 · 14) = 9 · 14 - 4 · 31 , 1 = (31 - 2 · 14) - (9 · 14 - 4 · 31) = 5 · 31 - 11 · 14 . Therefore - 11 · 14 = 1 mod 31 , x = - 6 · 11 = - 66 = 27 mod 31 . 2. (25 points) a)(15 points) Find all positive integers n such that n 2 + 3 n + 2 is a prime number.
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