I consider the first question. More generally, when is the -th power of an -cycle also an -cycle ?

We may assume that .

Consider the "orbit" (these are the images of 1 when is repeatedly applied; it may help to draw a sketch in some case, like n=8, k=2 and then k=3).

Saying that is an -cycle amounts to saying that (this is the cardinality of ).

The cardinality of is given by the first such that , which means that divides . Indeed (in case it is not clear), applying is like adding 1 modulo ; then applying is like adding modulo , and iff modulo . We have thus:

Now one can easily check that this quantity is equal to if, and only if and are relatively prime. (Or, more precisely ).

As a conclusion, the -th power of an -cycle is an -cycle if, and only if and are relatively prime.

It may not be the expected method (if , you may just look at every case individually), but I found it interesting to look at.

Laurent.