Find all functions mapping the positive integers to the positive integers such that $f^5(x)=kx$ for a positive integer $k$.
My approach to this problem consisted of trying out functions, as I cannot figure out a methodical way to approach such a problem. An example is when I tried functions such as $f(x)=x\sqrt[5]{k}$ and $f(x)=k/x$, these do not give positive integers for all values of $x$.
The case $k=1$ is relatively straightforward: $f^5(x)=x$, so each positive integer must be either a fixed point or part of a $5$-cycle. So that yields all (finite and infinite) products of disjoint $5$-cycles.
The case $k\gt1$ has an added twist. Imagine the positive integers arranged in towers of the form $rk^i$, one tower for each positive integer $r$ not divisible by $k$. Instead of numbers being connected in cycles, these towers need to be connected in cycles, and every iteration around the cycle needs to go up one step, in a sort of spiral. So you get all admissible functions by forming $5$-cycles from the positive integers not divisible by $k$ and picking a point in each cycle at which you multiply by $k$ (in addition to the cycling operation).
For instance, for $k=3$ you could have a cycle $(4,5,7,1,2)$, with the extra factor of $k$ inserted between $2$ and $4$, leading to the mappings
$$4\to5\to7\to1\to2\to12\to15\to21\to3\to6\to36\to\cdots\;.$$