Question

Give a proof of the theorem, "if n is odd, then n squared is odd".

Answer 1

Let $n$ be an odd number. Then $n=2k+1$ for some integer $k.$ It follows that $n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1=2s+1,$ where $s=2k^2+2k$ is an integer number. Therefore, $n$ squared is odd.