**Exercise 1.**
This software determines whether the two permutations \((1x)\) and \((123 ... n)\) generate \(S_n\) for various choices of \(x\) and \(n\). (That is, whether every element of \(S_n\) can be expressed as some product of these permutations. )
For \(n = 4\), run the program for \(x = 2, 3,\) and \(4\). For \(n = 5\), run the program for \(x = 2, 3, 4,\) and \(5\).
For \(n = 6\), run the program for \(x = 2, 3, 4, 5,\) and \(6\). For \(n = 8\), run the program for \(x = 2, 3, 4, 5, 6, 7,\) and \(8\). Conjecture a necessary and sufficient condition involving \(x\) and \(n\) for \((1x)\) and \((123 ... n)\) to generate \(S_n\).

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