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M135F08A8

# M135F08A8 - MATH 135 Assignment#8 Hand-In Problems 1 Solve...

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MATH 135 Fall 2008 Assignment #8 Due: Thursday 20 November 2008, 8:20 a.m. Hand-In Problems 1. Solve the simultaneous congruences 2 x 13 (mod 59) 5 x 42 (mod 34) 2. Solve the simultaneous congruences x 3 (mod 5) x 7 (mod 11) x 17 (mod 19) 3. Determine all solutions to the congruence x 13 + 7 x + 5 0 (mod 91). 4. If p and q are integers that are not divisible by 3 or 5, prove that p 4 q 4 (mod 15). 5. (a) Consider the system of simultaneous congruences x a (mod m ) x b (mod n ) where gcd( m, n ) = d . By following the technique for solving a system of congruences, prove that if this system has a solution, then d | a - b . (b) Consider the system of simultaneous congruences x 14 (mod 27) x 1 (mod 51) Prove that this system does not have a solution. (c) Consider the system of simultaneous congruences x 14 (mod 27) x 2 (mod 51) Determine the unique solution of this system modulo 27 · 51 gcd(27 , 51) . 6. In this problem, we determine the final two digits of the integer x = 753 132 .

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M135F08A8 - MATH 135 Assignment#8 Hand-In Problems 1 Solve...

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