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.