Let f be the function defined by
f(i,d) = (10 * i + d) mod n
, that is, the remainder of 10*i+d after division by n.
Read the number, digit by digit, remembering at every step the remainder r of the number after division by n.
We can do this by moving from r to f(r,d) on reading a digit d!
This means that we can do it using a finite state automaton.
For details and proof, see ariels's writeup on the case n=17.