q1 - 4. (3 points) Assume gcd( x,y ) = 1. Prove: gcd( x +...

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CMSC-37110 Discrete Mathematics FIRST QUIZ October 9, 2009 Name (print): Do not use book, notes, scratch paper. Show all your work. If you are not sure of the meaning of a problem, ask the instructor. The bonus problems are underrated, do not work on them until you are done with everything else. Write your solution in the space provided. You may continue on the reverse. This exam contributes 4% to your course grade. All variables in the problems below are integers . 1. (6 points) Prove by induction on k : ( x )( k 1)( if x is odd then x 2 k 1 (mod 2 k +2 )). 2. (2+3 points) (a) Define the greatest common divisors of a set A of integers. (b) Prove: gcd( ca,cb ) = | c | gcd( a,b ). Do not use unique prime factorization. State and use a basic fact about gcd we proved in class. (Recall that the gcd notation refers to the nonnegative number among the greatest common divisors.) 3. (2+4 points) (a) Define the prime property. (b) Use the preceding problem to show that every prime number has the prime property.
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Unformatted text preview: 4. (3 points) Assume gcd( x,y ) = 1. Prove: gcd( x + y,x-y ) = 1 or 2. 5. (2+4 points) True or false (circle one, prove). All quantiers range over the integers. (a) ( x )( y )(gcd( x,y ) = x-y ) T F (b) ( x )( y )( x 2-y 2 1 (mod 7)) T F 6. (3+4 points) Let a = 5 k + 1 and b = 3 k-2. Prove: (a) There exist innitely many values of k such that gcd( a,b ) 6 = 1. (b) If gcd( a,b ) 6 = 1 then gcd( a,b ) = 13. 7. (2+2+3+6B points) Compute each multiplicative inverse or prove it does not exist; your answer x (if exists) should be in the range 0 x < m where m is the modulus. k,x are positive integers. (a) 7-1 (mod 73) (b) 21-1 (mod 91) (c) k-1 (mod k 2 + k + 1) (d) BONUS: ( x + 1)-1 (mod x 2 + 1). 8. (8B points) BONUS. Let r,s > 0. Prove: gcd(2 s-1 , 2 t-1) = 2 d-1 where d = gcd( r,s )....
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This note was uploaded on 10/10/2009 for the course CMSC 37701 taught by Professor Xu during the Fall '09 term at UChicago.

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q1 - 4. (3 points) Assume gcd( x,y ) = 1. Prove: gcd( x +...

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