4023f11ex1a

4023f11ex1a - Name: Solutions Exam 1 Instructions. Answer...

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Unformatted text preview: Name: Solutions 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 . I Solution. Z / 24 = f 1 ; 5 ; 7 ; 11 ; 13 ; 17 ; 19 ; 23 g . J (b) How many elements are in the group Z / 1080 ? I Solution. j Z / 1080 j = (1080) = (2 3 3 3 5 = (2 3 ) (3 3 ) (5) = (2 3 2 2 )(3 3 3 2 )(5 1) = 4 18 4 = 288 : J (c) How many elements are in the group Z 12 Z 8 ? I Solution. j Z 12 Z 8 j = j Z 12 j j Z 8 j = 12 8 = 96. J 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 . I Solution. Use the matrix method for the Euclidean algorithm: 1 0 318 0 1 714 714 2 318=78 ! 1 318 2 1 78 318 4 78=6 ! 9 4 6 2 1 78 78 13 6=0 !...
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4023f11ex1a - Name: Solutions Exam 1 Instructions. Answer...

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