4023s09ex1

4023s09ex1 - Z 103 . Express your answer in the form [ b ]...

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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. [ 15 Points] Let A be a set with 5 elements, B a set with 7 elements, and C = f a; b; c g . Give the number of elements in each of the following sets. (a) The cartesian product A £ B £ C . (b) The power set P ( B ). Recall that P ( X ) is the set of all subsets of X . (c) The set P 2 ( B ) consisting of all 2-element subsets of B . (d) The set of all functions f : A ! C . (e) The set of all functions f : C ! A . 2. [20 Points] (a) Compute the greatest common divisor d = (2009 ; 1421) of the integers 2009 and 1421, and write d in the form d = 2009 ¢ s + 1421 ¢ t . (b) Compute the least common multiple m = [2009 ; 1421]. 3. [25 Points] This exercise makes use of the following equation: 1 = 7 ¢ 103 ¡ 48 ¢ 15 : Using this equation (i.e., it is not necessary to use the Euclidean algorithm to recreate it), answer the following questions. (a) Compute the multiplicative inverse of 15 in
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Unformatted text preview: Z 103 . Express your answer in the form [ b ] 103 where 0 b < 103. (b) Solve the congruence equation 15 x 8 mod 103. Express your answer in the form [ b ] 103 where 0 b < 103. (c) Solve the system of simultaneous linear congruences: x 8 mod 103 x 3 mod 48 : 4. [20 Points] This problem concerns arithmetic modulo 21. All answers should be expressed in the form [ a ] 21 with a an integer satisfying 0 a < 21. (a) Compute [9] 21 + [16] 2 21 . (b) Compute [9] 21 [16] 21 + [3] 21 [14] 21 . (c) Compute [5] 1 21 . (d) List the invertible elements of Z 21 . (e) List the zero divisors of Z 21 . 5. [20 Points] (a) State Eulers Theorem precisely. Be sure to carefully state the requisite hypotheses. (b) Compute (200). As usual ( n ) denotes the Euler -function applied to n . (c) Compute 7 177 mod 200. Math 4023 Section 1 February 11, 2009 1...
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