(a) Show that for all positive integers $z$ and $n$, either $z^{2 n} \equiv 0(\bmod 3)$ or $z^{2 n} \equiv 1(\bmod 3)$.

(b) If the positive integers $x, y, z$ satisfy $x^{2}+y^{2}=z^{2}$, show that at least one of $x$ and $y$ must be divisible by 3 . Can both $x$ and $y$ be odd?