Divisibility by Power of 10

Theorem
Let $r \in \Z_{\ge 1}$ be a strictly positive integer.

An integer $N$ expressed in decimal notation is divisible by $10^r$ the last $r$ digits of $N$ are all $0$.

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


 * $a_0 + a_1 10 + a_2 10^2 + \cdots + a_r 10^r = 0$.
 * $a_0 + a_1 10 + a_2 10^2 + \cdots + a_r 10^r = 0$.

Proof
Let $N$ be divisible by $10^r$.

Then:

Hence the result.