4023f11ex1

# 4023f11ex1 - (b) Compute 5 99 mod 21. 5. [14 Points] Let G...

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Name: Exam 1 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [ 12 Points] (a) List all of the elements in Z / 24 . (b) How many elements are in the group Z / 1080 ? (c) How many elements are in the group Z 12 £ Z 8 ? 2. [16 Points] (a) Use the Euclidean algorithm to compute d = gcd(318 ; 714). (b) Find integers s and t so that d = 318 ¢ s + 714 ¢ t . 3. [14 Points] (a) State the condition on a that is necessary and su–cient to insure that a has a multi- plicative inverse in Z n . (b) Compute the multiplicative inverse of 20 in Z 263 . Express your answer in the standard form of an integer b in the range 0 b < 263. 4. [14 Points] (a) What power m guarantees that the congruence a m · 1 mod 21 is valid for all integers a which are relatively prime to 21? Be sure to state which theorem you are using for your conclusion.
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Unformatted text preview: (b) Compute 5 99 mod 21. 5. [14 Points] Let G = Z / 16 = f 1 ; 3 ; 5 ; 7 ; 9 ; 11 ; 13 ; 15 g , where, as usual the group operation is multiplication modulo 16. Determine (with justication) if the following subsets of G are subgroups. (a) H = f 1 ; 3 ; 9 ; 11 g . (b) K = f 1 ; 5 ; 9 ; 11 g . 6. [14 Points] Answer the following questions concerning a cyclic group G = [ a ] with o ( a ) = 24. (a) What is the order j G j of G . (b) What is the order of the element a 20 . (c) How many subgroups of G are there? Be sure to include f e g and G in your count. (d) Is the group G Z 7 cyclic? Explain. 7. [16 Points] Find the smallest positive solution to the system of simultaneous linear congru-ences: x 9 mod 11 x 3 mod 9 : Math 4023 September 23, 2011 1...
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## This document was uploaded on 12/28/2011.

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