Fall2007 M2

Fall2007 M2 - z (mod m ). 4. Let a, b Z and m P . Suppose...

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MATH 135 Fall 2007 Midterm #2 Monday 12 November 2007, 7:00 p.m. to 8:15 p.m. 1. In each part of this problem, full marks will be given if the correct answer is written in the box. If your answer is incorrect, your work will be assessed for part marks. (a) Convert (2345) 6 to base 10. [3] (b) Convert (2007) 10 to base 12, using A and B to represent the digits 10 and 11, respectively. [3] (c) Determine the remainder when 2 34 56 52 + 3 19 is divided by 17. [3] (d) Determine the number of congruence classes in Z 18 that are solutions to the equation [3] [12][ x ] = [5]. 2. Suppose that q, u, w Z with q a prime number. Prove that if q | uw , then q | u or q | w . [4] 3. Suppose that x, y, z Z and m P . [4] Prove that if x y (mod m ) and y z (mod m ), then x
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Unformatted text preview: z (mod m ). 4. Let a, b Z and m P . Suppose that S is the statement: If a 0 (mod m ) or b 0 (mod m ), then ab 0 (mod m ). (a) Write down the converse of S . [2] (b) Prove or disprove the converse of S . [2] 5. Determine the complete solution to the linear congruence 46 x 14 (mod 270). [6] 6. (a) Solve the system of congruences [7] x 22 (mod 27) x 11 (mod 31) (b) If d = gcd( m, n ) and the system of congruences [4] x a (mod m ) x b (mod n ) has a solution x = x , prove that d | a-b . 7. Suppose that a is an integer with a 7 1 (mod 7). Prove that a 7 1 (mod 7 2 ). [4]...
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This note was uploaded on 10/21/2010 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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