M135F07A7

# M135F07A7 - MATH 135 Assignment#7 Hand-In Problems Fall...

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Fall 2007 Assignment #7 Due: Wednesday 07 November 2007, 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 x 2 (mod 11) 2. (a) Determine the inverse of [41] in Z 64 . (b) Determine the integer a with 0 a < 64 such that [ a ] = [13] - 1 [5] + [41] - 1 [9] in Z 64 . (c) Suppose that a is an odd integer and k is a positive integer. Explain why [ a ] has an inverse in Z 2 k . 3. Let p be an odd prime number. (a) Prove that x 2 + ab ( a + b ) x (mod p ) has exactly two solutions modulo p . (b) Solve the linear congruence 2 x 1 (mod p ). (c) Solve the congruence 4 x 3 x (mod p ). 4. Solve the following system of simultaneous equations in Z 13 . [3] [ x ] + [5] [ y ] = [7] [9] [ x ] + [6] [ y ] = [2] 5. Solve the simultaneous congruences 4 x 11 (mod 61) 7 x 21 (mod 30) 6. Solve the simultaneous congruences

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M135F07A7 - MATH 135 Assignment#7 Hand-In Problems Fall...

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