How to prove $n ≡ n_0 + n_1 + \dots +n_k \pmod{b-1}$ I am trying to prove this statement, where $n$ has base $b$ representation, which can be understood easily using this example:

In base $10$, mod $9$ of any number can be found by adding up its digits and doing the mod $9$ of that sum.

It's been a while since I've done proofs, and I'm just not sure where to start here. I know that, for example, in base $10$: 
$$10 \bmod 9 = 1\quad \text{or}\quad b \bmod b-1 = 1.$$
I believe I can substitute that in somehow, but I'm not sure how to start. Thanks for your help.

Hint: $$((b-1)+1)^k = 1 \quad (\operatorname{mod}\ b-1)$$
