**Exercise 5.**
For each positive integer \(n\), this software lists the number of elements of \(U(n)\) of each order. For each order \(d\) of some element of \(U(n)\), this software lists \(\phi(d)\) and the number of elements of order \(d\). (Recall that \(\phi(d)\) is the number of positive integers less than or equal to \(d\) and relatively prime to \(d\)). Do you see any relationship between the number of elements of order \(d\) and \(\phi(d)\)? Run the program for \(n =\) 3, 9 , 27, 81, 5, 25, 125, 7, 49, and 243. Make a conjecture about the number of elements of order \(d\) and \(\phi(d)\) when \(n\) is a power of an odd prime. Run the program for \(n =\) 6, 18, 54, 162, 10, 50, 250, 14, 98, and 686. Make a conjecture about the number of elements of order \(d\) and \(\phi(d)\) when \(n\) is twice a power of an odd prime. Make a conjecture about the number of elements of various orders in \(U(p^k)\) and \(U(2p^k)\) where \(p\) is an odd prime.

Please enter \(n\), the result will show below.