ECE-103-1041-Final_exam

# ECE-103-1041-Final_exam - 1 E&CE/C&O 103 FINAL EXAM WINTER...

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1 E&CE/C&O 103 FINAL EXAM WINTER 2004 QUESTION 1. [6] Let a n be defined by a 0 = 3 and, for n 1, a n = a n - 1 + 5. Prove n 0, a n (1 + 21) / 2. QUESTION 2: [7] Find all non-negative integer solutions to 21 x + 51 y = 15000. (If there are too many to list them all, list a few pairs and indicate somehow the range of the pairs.) QUESTION 3: (a) [3] Give a definition of “ a divides b ” (which we denote in symbols by a | b ). (b) [4] Use your definition to prove, from first principles, that if a | b and a | c , then, for any integers x and y , a | ( bx + cy ). QUESTION 4: [8] Solve the following system of two simultaneous congru- ences. 3 x 7 (mod 13) 4 x 9 (mod 19) QUESTION 4: (a) [5] Find the RSA public key corresponding to the pri- vate key (7 , 391). (It may help you to note that 391 = 17 × 23.) (b) [5] Use the RSA private key (7 , 391) to decode the message 281. QUESTION 6: Let S be the set of strings on the symbols 0 , 1 , 2 such that no 0 is immediately followed by a 1, no 1 is immediately followed by a 2, and no 2 is immediately followed by a 0. Let s n denote the number of strings in S of length n .

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