If $\displaystyle 8l(x^2+y^2+z^2+w^2)$, show that $\displaystyle x,y,z,w$ are even.
I know I have to classify squares mod 8, but can someone show me how.
If $\displaystyle a$ is even then $\displaystyle a=4k+1$ or $\displaystyle a=4k+3$. Notice $\displaystyle (4k+1)^2 = 16k^2 + 8k + 1 = 8(2k^2+k) + 1$. Also $\displaystyle (4k+3)^2 = 16k^2 + 24k + 9 = 8(2k^2+3k+1)+1$. Therefore, an odd integer modulo 8 is congruent to 1. Therefore, $\displaystyle x^2+y^2+z^2+w^2 \equiv 1,2,3,4 (\bmod 8)$ if at least one of them is odd. Thus, we require all four to be even.