Question

Prove that if x and y are both odd positive integers then x² + y² is even but not divisible by 4.

Answer 1

Let, n be any positive odd integer and let $x=n\mathrm{and}y=n+2$.
So, $x^2+y^2=(n{)}^2+(n+2{)}^2$
$\mathrm{Or},x^2+y^2=n^2+(n^2+4+4n)$
$$\begin{gathered} \Rightarrow x^2+y^2=2n^2+4+4n \\ \Rightarrow x^2+y^2=2(n^2+2+2n) \end{gathered}$$
$\Rightarrow x^2+y^2=2m$ (where $m=n^2+2n+2$)
Because $x^2+y^2$ has 2 as a factor, so the value is an even number.
Also, because it does not have any multiple of 4 as a factor, therefore, it is not divisible by 4.