Divisibility by 2

Theorem
An integer $N$ expressed in decimal notation is divisible by $2$ the  of $N$ is divisible by $2$.

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 $2$


 * $a_0$ is divisible by $2$.
 * $a_0$ is divisible by $2$.

Proof
Let $N$ be divisible by $2$.

Then: