Divisibility by 9

Theorem
A number:
 * $N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$

is divisible by $9$ iff the sum:
 * $a_0 + a_1 + \ldots + a_n$

of its digits is divisible by $9$.

Or, commonly stated, a number is divisible by $9$ if and only if the sum of its digits is divisible by $9$.

Direct Proof
If $N$ is divisible by 9, then

Alternative Proof
It can be seen that this is a special case of Congruence of Sum of Digits to Base Less 1.

Divisibility by 3
This same argument holds for divisibility by $3$. The proof is exactly the same as the proof above, replacing all instances of $9$ by $3$.