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ECE-103-1051-Final_exam

# ECE-103-1051-Final_exam - 1 FINAL EXAMINATION WINTER TERM...

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1 FINAL EXAMINATION WINTER TERM 2004 E&CE/C&O 103 QUESTION 1. (a) [2 points] Show that if n 1 is an integer, then 3 n + 3 n + 1 = 3 (3 n + 2)(3 n + 1) (2 n + 2)(2 n + 1) 3 n n . [4] Prove that, for every integer n 1, 3 n n 1 2 3 n 3 2 2 n . QUESTION 2: [7] Find all solutions to 25 x + 60 y = 15. QUESTION 3: (a) [3] Give a definition of “greatest common divisor” of positive integers a and b . (b) [4] Assume a , b and d are positive integers such that d | a and d | b . Show that if there are integers x, y such that d = ax + by , then d is the greatest common divisor of a and b . QUESTION 4: [8] Solve the following system of two simultaneous congru- ences. x 7 (mod 14) x 9 (mod 25) QUESTION 4: (a) [5] Find the RSA public key corresponding to the pri- vate key (5 , 397). (It may help you to note that 397 = 13 × 29.) (b) [5] Use the RSA private key (5 , 397) to decode the message 281. QUESTION 6: Let S denote the set of strings of 0’s and 1’s such that no four consecutive terms are the same (either all 0’s or all 1’s). For example, 00111001 ∈ S , while 001111001 / ∈ S . Let s n denote the number of strings in S that have length n . The following are the fourteen strings in S that have length 4: 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 (a) [3 points] For i = 1 , 2 , 3, let s n,i

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ECE-103-1051-Final_exam - 1 FINAL EXAMINATION WINTER TERM...

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