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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.
- Fall '09