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Exercise 2. This software computes the elements of the

ISBN: 9781133599708 52

Solution for problem 2CE Chapter 8

Contemporary Abstract Algebra | 8th Edition

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Contemporary Abstract Algebra | 8th Edition

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Problem 2CE Problem 2CE

Exercise 2. This software computes the elements of the subgroup U(n)k = {xk | x U(n)} of U(n) and its order. Run the program for (n,k) = (27,3), (27,5), (27,7), and (27,11). Do you see a relationship connecting|U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (25,3), (25,5), (25,7), and (25,11). Do you see a relationship connecting |U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (32,2), (32,4), and (32,8). Is your conjecture valid for U(32,16)? If not, restrict your conjecture. Run the program for (n,k) = (77,2), (77,3), (77,5), (77,6), (77,10), and (77,15)? Do you see a relationship among U(77,6) and U(77,2), and U(77,3)? What about U(77,10), U(77,2), and U(77,5)? What about U(77,15), U(77,3), and U(77,5)? Make a conjecture. Use the theory developed in this chapter about expressing U(n) as external direct products of cyclic groups of the form Zn to analyze these groups to verify your conjectures.

Exercise 3. This software implements the algorithm given in Chapter 8 to express U(n) as an external direct product of groups of the form Zk. Assume that n is given in prime-power factorization form. Run your program for 3 . 5 . 7, 16 . 9 . 5, 8 . 3 . 25, 9 . 5 . 11, and 2 . 27 . 125. [ NOTE: Please enter the prime-power factorization form with a period(".")' in between the integers and without any space. Also, this program has been written to accept n as any integer, i.e., instead of entering n in the factored form as 3 . 5 . 7 you could enter 105 . ]

Exercise 5. This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of Mr mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input.

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Chapter 10: Pure Competition in the Short Run  Four market models o Pure competition- involves a very large number of firms producing a standardized product  “Price taker”- it cannot change market price; it can only adjust to it  Free entry and exit o Pure monopoly- a market structure in which one firm is the sole seller of a product or service o Monopolistic competition- characterized by a relatively large number of sellers producing differentiated products o Oligopoly- involves only a few sellers of a standardized or differentiated product  Purely competitive demand o Perfectly elastic demand  Firm produces as much or as little as they wish at the

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ISBN: 9781133599708

This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. This full solution covers the following key subjects: program, conjecture, form, run, Relationship. This expansive textbook survival guide covers 34 chapters, and 2038 solutions. The full step-by-step solution to problem: 2CE from chapter: 8 was answered by , our top Math solution expert on 07/25/17, 05:55AM. Since the solution to 2CE from 8 chapter was answered, more than 341 students have viewed the full step-by-step answer. The answer to “Exercise 2. This software computes the elements of the subgroup U(n)k = {xk | x U(n)} of U(n) and its order. Run the program for (n,k) = (27,3), (27,5), (27,7), and (27,11). Do you see a relationship connecting|U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (25,3), (25,5), (25,7), and (25,11). Do you see a relationship connecting |U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (32,2), (32,4), and (32,8). Is your conjecture valid for U(32,16)? If not, restrict your conjecture. Run the program for (n,k) = (77,2), (77,3), (77,5), (77,6), (77,10), and (77,15)? Do you see a relationship among U(77,6) and U(77,2), and U(77,3)? What about U(77,10), U(77,2), and U(77,5)? What about U(77,15), U(77,3), and U(77,5)? Make a conjecture. Use the theory developed in this chapter about expressing U(n) as external direct products of cyclic groups of the form Zn to analyze these groups to verify your conjectures.Exercise 3. This software implements the algorithm given in Chapter 8 to express U(n) as an external direct product of groups of the form Zk. Assume that n is given in prime-power factorization form. Run your program for 3 . 5 . 7, 16 . 9 . 5, 8 . 3 . 25, 9 . 5 . 11, and 2 . 27 . 125. [ NOTE: Please enter the prime-power factorization form with a period(".")' in between the integers and without any space. Also, this program has been written to accept n as any integer, i.e., instead of entering n in the factored form as 3 . 5 . 7 you could enter 105 . ]Exercise 5. This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of Mr mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input.” is broken down into a number of easy to follow steps, and 357 words. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708.

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