Question

Prove that if $x\mathrm{and}y$ are both odd positive integers, then $x^2+y^2$ is even but not divisible by $4$.

Answer 1

Let the two odd positive numbers $x\mathrm{and}y$ be $2k+1$ and $2p+1$ respectively

So, $x^2+y^2={(2k+1)}^2+{(2p+1)}^2$

$=4k^2+4k+1+4p^2+4p+1$

$=4k^2+4p^2+4k+4p+2$

$=4(k^2+p^2+k+p)+2$

So, the sum of the square is even then number is not divisible by $4$.

Therefore, if $x\mathrm{and}y$ are odd positive integers, then $x^2+y^2$ is even but not divisible by $4$.

Hence, Proved.