For each positive integerExercise 4.n, this software gives the order ofU(n)and the order of each element inU(n). Do you see any relationship between the order ofU(n)and the order of its elements? Run the program forn =8, 16, 32, 64 and 128. Make a conjecture about the number of elements of order2inU(2when^{k})kis at least3. Make a conjecture about the number of elements of order4inU(2when^{k})kis at least4. Make a conjecture about the number of elements of order8inU(2when^{k})kis at least5. Make a conjecture about the maximum order of any element inU(2when^{k})kis at least3. Try to find a formula for an element of order4inU(2when^{k})kis at least4.

For each positive integerExercise 5.n, this software lists the number of elements ofU(n)of each order. For each orderdof some element ofU(n), this software listsphi(d)and the number of elements of orderd. (Recall thatphi(d)is the number of positive integers less than or equal todand relatively prime tod). Do you see any relationship between the number of elements of orderdandphi(d)? Run the program forn = 3, 9 , 27, 81, 5, 25, 125, 7, 49,and243. Make a conjecture about the number of elements of orderdandphi(d)whennis a power of an odd prime. Run the program forn = 6, 18, 54, 162, 10, 50, 250, 14, 98,and686. Make a conjecture about the number of elements of orderdandphi(d)whennis twice a power of an odd prime. Make a conjecture about the number of elements of various orders inU(p) and^{k}U(2pwhere^{k})pis an odd prime.

For each positive integerExercise 6.n, this software gives the order ofU(n). Run the program forn = 9, 27, 81,and243. Try to guess a formula for the order ofU(3when^{k})kis at least2. Run the program forn = 18, 54, 162,and486. How does the order ofU(2x3appear to be related to the order of^{k})U(3? Run the program for^{k})n = 25, 125,and625. Try to guess a formula for the order ofU(5when^{k})kis at least2. Run the program forn = 50, 250,and1250. How does the order ofU(2x5appear to be related to the order of^{k})U(5? Run the program for^{k})n = 49and343. Try to guess a formula for the order ofU(7when^{k})kis at least2. Run the program forn = 98and686. How does the order ofU(2x7appear to be related to the order of^{k})U(7)? Based on your guesses for^{k}U(3guess a formula for the order of^{k}),U(5^{k}) and U(7^{k})U(pwhen^{k})pis an odd prime andkis at least2. What about the order ofU(2xpwhen^{k})pis an odd prime andkis at least2. Does your formula also work whenkis1?

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