Question

Show that the square of an even number is an even number using a direct proof.

Answer 1

Given $n$ is even. Then $\exist k\in \Z$ such that $n=2k.$

Then

$$n^2=(2k)^2=4k^2=2(2k^2)$$

Let $m=2k^2, k\in \Z.$ Then $m=2k^2, m\in \Z.$

So $n^2=2m, m\in \Z.$ This means that $n^2$ is even.

Therefore the square of an even number is an even number.