M135W09A7 - Z 19 [5][ x ] + [4][ y ] = [3] [4][ x ] + [3][...

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MATH 135 Winter 2009 Assignment #7 Due: Wednesday 11 March 2009, 8:20 a.m. Hand-In Problems 1. In each part, determine if the congruence has solutions. If it does, determine the complete solution. (a) 1653 x 77 (mod 2000) (b) 1492 x 77 (mod 2000) (c) x 2 4 x (mod 12) (d) x 11 + 3 x 10 + 5 2 (mod 11) 2. Let p be an odd prime and let a and b be integers. (a) Prove that the congruence x 2 - ab ( a - b ) x (mod p ) always has at least one solution. (b) When does the above congruence have 2 solutions modulo p ? (c) Solve the congruence 4 x 3 x (mod p ). 3. Solve the following system of equations in
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Unformatted text preview: Z 19 [5][ x ] + [4][ y ] = [3] [4][ x ] + [3][ y ] = [17] . 4. Solve the following system of congruences equations 2 x + 4 y 2 (mod 14) 37 x-5 y 1 (mod 14) 5. Calculate [13]-1 [5] + [41]-1 [9] in Z 64 . 6. Prove that if [ a ] has an inverse in Z m , then so does [ a n ] (for all positive integers n ) and [ a n ]-1 = ([ a ]-1 ) n . Recommended Problems 1. Text, page 83, #46 2. Text, page 83, #47 3. Text, page 83, #31 4. Text, page 84, #62 5. Text, page 84, #63...
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