**a. The elements of***U(n)*.**b. The inverse of each member of***U(n)*.**Run your program for***n*= 12, 15, 30, 36, 63.

Exercise 2.This software determines the size ofU(k). Run the program fork = 9, 27, 81, 243, 25, 125, 49, and 121. On the basis of this output try to guess a formula for the size ofU(pas a function of the prime p and the integer n. Run the program for^{n})k = 18, 54, 162, 486, 50, 250, 98,and 242. Make a conjecture about the relationship between the size ofU(2pand the size of^{n})U(pwhere^{n})pis a prime greater than2.

This software computes the inverse of any element inExercise 3.GL(2,Z, where_{p})pis a prime.

This software determines the number of elements inExercise 4.GL(2and,Z_{p})SL(2,Zwhere_{p})pis a prime. (The technical term for the number of elements in a group is theorderof the group.) Run the program forp= 3, 5, 7, and 11. Do you see a relationship between the orders ofGL(2,Zand_{p})SL(2,Zand_{p})p-1 ? Does this relationship hold forp=2 ? Based on these examples does it appear that p always divides the order ofSL(2,Z? What about_{p})p-1 ? What aboutp+1 ? Guess a formula for the order ofSL(2,Z. Guess a formula for_{p})GL(2,Z._{p})