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Unformatted text preview: D 0 in Z n . 1.7.14. Suppose a is relatively prime to n . (a) Show that for all b 2 Z , the congruence ax b . mod n/ has a solution. (b) Can you nd an algorithm for solving congruences of this type? Hint: Consider Exercise 1.7.11 . (c) Solve the congruence 8x 12 . mod 125/ . 1.7.15. This exercise guides you to a proof of the Chinese remainder theorem. (a) To prove the Chinese remainder theorem, show that it sufces to nd m and n such that C ma D C nb . (b) To nd m and n as in part (a), show that it sufces to nd s and t such that as C bt D . / . (c) Show that the existence of s and t as in part (b) follows from a and b being relatively prime. 1.7.16. Find and integer x such that x 3 . mod 4/ and x 5 . mod 9/ . 1.8. Polynomials Let K denote the set Q of rational numbers, the set R of real numbers, or the set C of complex numbers. ( K could actually be any eld ; elds are...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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