Yes, it's easy: to test divisibility by n make a graph with n verities labeled 0 through n-1; put black arrows from each vertex k to (k+1) mod n; and put white

Yes, it's easy: to test divisibility by n make a graph with n verities labeled 0 through n-1; put black arrows from each vertex k to (k+1) mod n; and put white arrows from each k to 10k mod n. (That's for testing divisibility by n of numbers written in base 10; for others bases just replace the 10 above.)