Here is a purely number theoretical question that I got to know from our electrical engineering department.

Call a number $q\in \mathbb{N}$, good if one can do the following:

Given a set of "probabilistic" switches, each of which is open with probability $\frac{a}{q}$, $a=1,2,\dots,q-1$, and two nodes $U,V$ one can build a simple series parallel circuit (where one can use each type of switch more than once) connecting $U$ to $V$ where the probability of $U\to V$ being open is exactly $\frac{b}{q^n}$ for any $n,b\in \mathbb{N}$ such that $b\le q^n-1$.

The question is which numbers are good? I think the conjecture is that only numbers which are multiples of $2$ or $3$ are good. $5$ for example is not good as one can not construct a circuit which is open with probability exactly $\frac{7}{25}$.

P.S. A "simple series parallel" circuit is one that can be build recursively by the operation of placing a switch in series with our circuit or placing a switch in parallel with our circuit. For example the wheatstone bridge is not simple series parallel.