Divisibility by 8

Theorem
An integer $N$ expressed in decimal notation is divisible by $8$ the $3$ s of $N$ form a $3$-digit integer divisible by $8$.

That is:
 * $N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $8$


 * $100 a_2 + 10 a_1 + a_0$ is divisible by $8$.
 * $100 a_2 + 10 a_1 + a_0$ is divisible by $8$.

Proof
Let $N$ be divisible by $8$.

Then: