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Mathematics 366 Points (leave blank) 1 2 3 4 Midterm 1 (100 points) 5 ∑ 10/4/13 Name: (clearly, please) You may use a pocket calculator that does not have internet access. You can work on the problems in any order you like. Show your work! All problems carry the same weight. Calculation steps and explanation for statements made are a crucial part of a solution. Partial credit will be given sparingly – rather complete one problem than start two only partially. 1) a) Find an integer solution x for the equation 13x mod 82 = 1 b) Let p = 83 which is prime. Then 213 ≡ 58 (mod 83). Find a number e such that 58e ≡ 2 (mod 83). (Hint: 83 − 1 = 82, use part a)) 2) How many numbers are between 0 and 10000 (inclusive) which contain at least one even and one odd digit? 3) Show, usings Fermats little theorem, that n = 133 is not prime. (No points will be given for showing it by giving a factorization of n.) You may use that n − 1 = 27 + 22 . Explain your calculations. 4) Decrypt the message KONWF RNSEA ZSJ which has been encrypted using a Vigenè€re cipher with keyword RAMS. You will find a tabula recta on the last page of this exam. 5) Let a be a primitive root modulo p. (That is, OrderMod p (a) = p − 1.) Show that for every 1 ≤ b < p there exists an exponent e such that a e ≡ b (mod p). (Hint: Show that for 0 ≤ e < f < p −1 you have that a e ≡/ a f (mod p), i.e. all powers are different. Then use the pidgeonhole principle.)