Question

Use a direct proof to show that the sum of two odd integers is even

Answer 1

If the integer $n$ is odd, then $\exist k\in \Z$ such that $n=2k+1.$

The sum of two odd integers we can write as

$$n_1+n_2=2k_1+1+2k_2+1$$

$$=2(k_1+k_2+1), k_1, k_2\in \Z$$

Let $m=k_1+k_2+1, k_1, k_2\in \Z.$ Then $m\in \Z.$

We see that $\exist m\in \Z$ such that $n_1+n_2=2m.$

This means that the sum $n_1+n_2$ is an even number.

Therefore the sum of two odd integers is even.