**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.