Non inductive proof for square of odd integers

Question

Can we argue that the square of every odd integer is of the form $8k+1$ using non inductive proof?

Answer 1

Sure. If $n=2k+1$ then $n^2=4k^2+4k+1=4k(k+1)+1$ and either $k$ or $k+1$ is even, so $4k(k+1)$ is divisible by $8$.