# Divisibility by 2

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## 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 $10 \equiv 0 \pmod 2$ |

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$