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 $2$

if and only if:

$a_0$ is divisible by $2$.

Let $N$ be divisible by $2$.

Then:

$\ds N$

$\equiv$

$\ds 0 \pmod 2$

$\ds \leadstoandfrom \ \ $

$\ds \sum_{k \mathop = 0}^n a_k 10^k$

$\equiv$

$\ds 0 \pmod 2$

$\ds \leadstoandfrom \ \ $

$\ds a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}$

$\equiv$

$\ds 0 \pmod 2$

$\ds \leadstoandfrom \ \ $

$\ds a_0$

$\equiv$

$\ds 0 \pmod 2$

as $10 \equiv 0 \pmod 2$

$\blacksquare$

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisible
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisible