Fall2006 M2

Fall2006 M2 - Z b a , and P m with m > 1. (a) (1 mark)...

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1. (6 × 3 marks) In each part of this problem, full marks will be given if the correct answer is written in the (imaginary) box. If your answer is incorrect, your work will be assessed for part marks. (a) Convert (4321) 5 to base 10. (b) Convert (2345) 10 to base 13, using A as the digit representing 10, B as the digit representing 11, and C as the digit representing 12. (c) Calculate lcm(2 3 3 4 5 6 , 1600). (d) Determine the digit d so that (2 d 4 000 000 234 458) 10 is divisible by 9. (e) Determine the remainder when 2 203 + 5 300 7 100 is divided by 7. (f) Determine [5] -1 in Z 32 . 2. (5 marks) Suppose that q is a prime number and Z y x , . Prove that if q | xy , then q | x or q | y . 3. Suppose
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Unformatted text preview: Z b a , and P m with m > 1. (a) (1 mark) Give the definition of the statement ( ) m b a mod . (b) (4 marks) Prove that if a and b have the same remainder when divided by m , then ( ) m b a mod . 4. (8 marks) Solve the simultaneous congruencies ( ) ( ) 64 mod 59 81 mod 32 x x 5. (3 marks) Consider the statement For all Z c b a , , , if a | bx + cy for all integers x and y , then a | b and a | c . Is this statement TRUE or FALSE? Prove your answer. 6. (6 marks) Suppose that p , q and r are prime numbers and that p is odd. If p | 2 q + r and p | 2 q r , prove that q = r ....
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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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