Prove the language of numbers divisible by 3 is not regular using pumping lemma
You cannot choose any string for the pumping lemma. You have to choose one that is at least as long as the pumping distance of the language (which exists when it is regular), but you do not know which it is, so you have to assume it is some value named $p$ (or some other name) and you assume you have a well chosen string in the language that is greater than $p$.

Another point is that I would say that $\epsilon$ is not the representation of a decimal number, rather than say it is not divisible by $3$.

All this being said, modular arithmetics works pretty well with divisibility constraints, divisibility by a constant, I mean. And Finite Automata are usually friendly to modular arithmetics because it is defined on finite sets of values (or classes of numbers).