Divisibility by 9

Theorem
A number expressed in decimal notation is divisible by $9$ the sum of its digits is divisible by $9$.

That is:
 * $N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$


 * $a_0 + a_1 + \ldots + a_n$ is divisible by $9$.
 * $a_0 + a_1 + \ldots + a_n$ is divisible by $9$.