Prove that, if x and y are both odd positive integers,

Let x = 2m + 1 and y = 2m + 3 are odd positive integers, for every positive integer m.

Then, $\quad x^{2}+y^{2}=(2 m+1)^{2}+(2 m+3)^{2}$

$=4 m^{2}+1+4 m+4 m^{2}+9+12 m \quad\left[\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$

$=8 m^{2}+16 m+10=$ even

$=2\left(4 m^{2}+8 m+5\right)$ or $4\left(2 m^{2}+4 m+2\right)+1$

Hence, $x^{2}+^{\prime} y^{2}$ is even for every positive integer $m$ but not divisible by 4 .