How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$?
For this one the answer given is the $2^{16}$, but how could we prove that that this is the only possible solution? and what about the general case of $x^n, \text{ with } x,n \in \mathbb{N}$?