In Strauss's Partial Differential Equations, the eigenvalue problem $$-\Delta v=\lambda v,\qquad v\big|_{\partial \Omega}=0$$ is solved by separating the $x,y,z$ variables: $v=X(x)Y(y)Z(z)$, $$ \frac{X''}{X}+\frac{Y''}{Y}+\frac{Z''}{Z}=-\lambda$$
The separated BCs are $$X(0)=X(\pi)=Y(0)=Y(\pi)=Z(0)=Z(\pi)=0$$
Here are my questions:
How do we deduce that the solutions are $$v(x,y,z)=\sin lx\sin my \sin nz$$ where $$l^2+m^2+n^2=\lambda\quad (1\leq m,l,n<\infty)$$
Why the shape of $\Omega=[a,b]\times[c,d]\times[e,f]$ is needed for this kind of method?