# Divisibility by 2

## Theorem

An integer $N$ expressed in decimal notation is divisible by $2$ if and only if the least significant digit 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 $2$.

## Proof

Let $N$ be divisible by $2$.

Then:

 $\displaystyle N$ $\equiv$ $\displaystyle 0 \pmod 2$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \sum_{k \mathop = 0}^n a_k 10^k$ $\equiv$ $\displaystyle 0 \pmod 2$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}$ $\equiv$ $\displaystyle 0 \pmod 2$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a_0$ $\equiv$ $\displaystyle 0 \pmod 2$ as $10 \equiv 0 \pmod 2$

$\blacksquare$