Question

Prove that if $x$ and $y$ are odd positive integer, then $x^2+y^2$ is even but not divisible by $4.$

Answer 1

We know that any odd positive integer is of the form $2q+1$ for some integer $q$. So, let $x=2m+1$ and $y=2n+1$ for some integer $m$ and $n.$

$\therefore x^2+y^2=(2m+1{)}^2+(2n+1{)}^2$
$\Rightarrow x^2+y^2=4(m^2+n^2)+4(m+n)+2$

$\Rightarrow x^2+y^2=4\left\{\right(m^2+n^2)+(m+n\left)\right\}+2$
$\Rightarrow x^2+y^2=4q+2$, where $q=(m^2+n^2)+(m+n)$

$\Rightarrow x^2+y^2$ is even and leaves remember $2$ when divided by $4$
$\Rightarrow x^2+y^2$ is even but not divisible by $4$