135-finalexaminformation-2011

# 135-finalexaminformation-2011 - Math 135 Final Exam...

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Math 135 Final Exam Information Time Tuesday, April 12, 2011, 4:00-6:30pm Location check https://odyssey.uwaterloo.ca/teaching/schedule Materials all materials covered in class and in Assignment 1-11. Short list of proofs (you need to know the proofs of the following results) Proposition 2.27, Proposition 2.28, Proposition 2.51, Theorem 2.52, Proposition 3.14, Theorem 3.42, Theorem 7.41, Theorem 8.61 Practise Exam (Math 135 Final Exam Fall 2010) (recommended time: 150 minutes) 1 Find all possible combinations of 21-cent and 57-cent stamps which can be used to mail a package which requires postage of exactly 9 dollars. 2 Find gcd(50! , 2 50 ). 3 Prove that there are infinitely many primes. 4 (a) Find all solutions of the following system of linear congruences in Z 10 [7][ x ] + [9][ y ] = [5] [5][ x ] + [3][ y ] = [7] , (b) Determine all solutions of the congruence equation x 61 + 26 x 41 + 11 x 25 + 5 = 0 (mod 143). 5 (a) Compute φ (100), where φ is the Euler Phi function. (b) Find the last two digits of 4153 (41 53 ) . 6 (a) Use the square and multiply algorithm to encrypt the message M = 7 using RSA public key ( e, n ) = (17 , 551).

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