4023f10ex1

4023f10ex1 - + [3] 22 [14] 22 . (c) Compute [5] 1 22 . (d)...

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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 T = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g . Give the number of elements in each of the following sets. (a) The cartesian product T £ T . (b) The power set P ( T ). Recall that P ( X ) is the set of all subsets of X . (c) The set P 3 ( T ) consisting of all 3-element subsets of T . (d) P ( T ) [ T . (e) The set of all functions f : T ! B , where B = f 0 ; 1 g is the Boolean set. 2. [18 Points] (a) Compute the greatest common divisor d = (2010 ; 5159) of the integers 2010 and 5159, and write d in the form d = 2010 ¢ s + 5159 ¢ t . (b) Compute the least common multiple m = [2010 ; 5159]. 3. [15 Points] This problem concerns arithmetic modulo 22. All answers should be expressed in the form [ a ] 22 with a an integer satisfying 0 a < 22. (a) Compute [4] 2 22 + [11] 22 . (b) Compute [9] 22 [8] 22
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Unformatted text preview: + [3] 22 [14] 22 . (c) Compute [5] 1 22 . (d) List the units of Z 22 . (e) List the zero divisors of Z 22 . 4. [18 Points] Find all solutions to the following congruence equation: 35 x 20 (mod 130) : 5. [18 Points] Find the smallest positive solution to the system of simultaneous linear congru-ences: x 4 mod 110 x 3 mod 63 : 6. [16 Points] Recall that ( n ) denotes the Euler -function applied to n . (a) Give the denition of ( n ). That is, what (precisely) does ( n ) count? (b) State Eulers Theorem concerning the powers of a modulo n precisely. Be sure to care-fully state the requisite hypotheses. (c) Compute (675). Note that 675 = 27 25. (d) Use Eulers Theorem to compute 2 368 (mod 675). Math 4023 September 22, 2010 1...
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