Exercise 2. Let Z_n[i] = { a+bi | a, b belong to Z_n, i²=-1 } (the Gaussian integers modulo n ). This software finds the group of units of this ring and the order of each element of the group. Run the program for n = 3, 7, 11, and 23. Is the group of units cyclic for these cases? Try to guess a formula for the order of the group of units of Z_n[i] as a function of n when n is a prime and n mod 4 = 3. Run the program for n = 9 and 27. Are the groups cyclic? Try to guess a formula for the order when n = 3^k. Run the program for n = 5, 13, 17, and 29. Is the group cyclic for these cases? What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of Z_n[i] as a function of n when n is a prime and n mod 4 = 1. Try to guess a formula for the largest order of any element in the group of units of Z_n[i] as a function of n when n is a prime and n mod 4 = 1. On the basis of the orders of the elements of the group of units, try to guess the isomorphism class of the group. Run the program for n = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the isomorphism class of the group.