Exercise 6. For each positive integer $$n$$, this software gives the order of $$U(n)$$. Run the program for $$n = 9, 27, 81,$$ and $$243$$. Try to guess a formula for the order of $$U(3k)$$ when $$k$$ is at least $$2$$. Run the program for $$n = 18, 54, 162,$$ and $$486$$. How does the order of $$U(2\times3k)$$ appear to be related to the order of $$U(3k)$$? Run the program for $$n = 25, 125$$, and $$625$$. Try to guess a formula for the order of $$U(5k)$$ when $$k$$ is at least $$2$$. Run the program for $$n = 50, 250,$$ and $$1250$$. How does the order of $$U(2\times5k)$$ appear to be related to the order of $$U(5k)$$? Run the program for $$n = 49$$ and $$343$$. Try to guess a formula for the order of $$U(7k)$$ when $$k$$ is at least $$2$$. Run the program for $$n = 98$$ and $$686$$. How does the order of $$U(2\times 7k)$$ appear to be related to the order of $$U(7k)$$? Based on your guesses for $$U(3k)$$, $$U(5k)$$ and $$U(7k)$$ guess a formula for the order of $$U(pk)$$ when $$p$$ is an odd prime and $$k$$ is at least $$2$$. What about the order of $$U(2\times pk)$$ when $$p$$ is an odd prime and $$k$$ is at least $$2$$. Does your formula also work when $$k$$ is $$1$$ ?

Please enter $$n$$, the result will show below.