Answer to Question #349806 in Discrete Mathematics for janine

Question

Use a direct proof to show that the sum of two odd integers(k and l) is even.

Answer 1

Proof.

Let $k$ and $l$ be the odd integers. Then there exist some integers $k_0$ and $l_0$ such that

$k=2k_0+1,$

$l=2l_0+1.$

We can find the sum:

$k+l=(2k_0+1)+(2l_0+1)=2k_0+2l_0+2=2(k_0+l_0+1),$

where $k_0+l_0+1$ is an integer as a sum of integers.

So $k+l$ is divisible by 2 and is even.

The proof is finished.