Question

Prove that, if x and y are odd positive integers, 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 $2q+1$, where $q$ is an integer.

So, let $x=2m+1$ and $y=2n+1$, for some integers m and n.

we have $x^2+y^2$

$x^2+y^2=(2m+1{)}^2+(2n+1{)}^2$

$x^2+y^2=4m^2+1+4m+4n^2+1+4n=4m^2+4n^2+4m+4n+2$

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

$x^2+y2=4q+2$, when $q=(m^2+n^2)+(m+n)$

$x^2+y^2$ is even and leaves remainder 2 when divided by 4.

$x^2+y^2$ is even but not divisible by 4.