Problem:Consider positive integers $s,t$. Does there exist a function $f$ from $\mathbb{N}$ onto $\mathbb{N}$ such that $f^{s}(n)=tn$ for all $n$ where
$f^{s}(n)=f(f(f(..f(n))))\text{(s compositions of f)}$
So here is My attempt, I want some one to check.
$\textbf{Solution:}$
Easy to note $f$ must be bijective
$\textbf{Case 1:}$ Suppose there exist a fixed point then $t=1$ (forced)
$\textbf{Case 2:}$ Else if there must be a cycle for $f(1),f^{2}(1),....f^{k}(1)=1$ for some postive integer $k$ say that $k$ is least such then it is easy to note $f^{ks}(1)=1$ but $f^{s}(n)=tn$ thus $f^{s+r}(n)=tf^{r}(n)$ for all non-negative integer $r$ thus we have $t|f^{d}(n)$ for all $n\geq 1$ and for all $d\geq s$ hence choosing $d=ks$ and $n=1$ we have $t|1$ then $t=1$!!!
Hence in both case $t=1$ this we need to look $f^{s}(n)=n$ but then we can easy choose $f(x)=x$
$\textbf{Conclusion:}$
$f(x)=x$ is such function whenever $t=1$ unless If $t>1$ we cannot have Case 1 and 2 to be true.