Divisibility by 10

Theorem
An integer $N$ expressed in decimal notation is divisible by $10$ the least significant digit of $N$ is $0$.

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 $10$


 * $a_0 = 0$
 * $a_0 = 0$

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

Then: