Prove that if the sum of digits of a number is divisible by 3, so is the number itself. Here is the proof of the converse:
Iff a number $n$ is divisible by $3$, then the sum of its digits is also divisible by $3$.
Proof:
We know $n \mod 3 = 0$. By the basis representation theorem, $n$ can be re-written as $n_k10^k + n_{k-1}10^{k-1} + \cdots + 10n_1 + n_0 \equiv 0 \mod 3$. By the modular arithmetic addition rule, we can take $n_k10^k \mod 3 + n_{k-1}10^{k-1} \mod 3 + \cdots + 10n_1 \mod 3 + n_0 \mod 3$. But then we just get $n_k + n_{k-1} + \cdots + n_1 + n_0$, since $10 \mod 3 \equiv 1$. 
Now suppose that the sum of the digits was divisible by 3. How can I prove that the representation in base $10$ is divisible by $3$?

I hope this will suffice:
$$
n
= (d_{m-1}\cdots d_0)_{10} 
= \sum_{k=0}^{m-1} d_k 10^k
= \sum_{k=0}^{m-1} d_k (10^k - 1) + d_k 
= \sum_{k=0}^{m-1} d_k (10^k - 1) + S(n)
$$
with $S(n)$ being the digit sum in base $10$ and
$$
10^k - 1 = 9 \frac{10^{k} - 1}{10-1} = 9 \sum_{i=0}^{k-1} 10^i 
= \sum_{i=0}^{k-1} 9 \, 10^i
= (\underbrace{9\cdots 9}_{k \times})_{10} 
$$
is divisible by $3$, it depends on the divisibility of $S(n)$.
The same statement should hold for division by $9$.
