I have posted an article on my blog about the mathematics behind digital roots. This is an extension and amplification of the concepts that were first introduced in my Vedic Mathematics post devoted to digital roots.
Most people know digital roots as the remainders of division by 9. But that is just one of the countless possible digital roots. Remainders after division by any number can be utilized as digital roots, especially for verification of the results of arithmetic operations. And some of those digital roots are much more useful than digital roots with respect to 9. You can find out all the details, as well as suggestions for useful digital roots here.