 Divisor class groups of products of projective spaces

For positive $n,m$, $\mathrm{Cl}({\mathbb{P}}^n\times {\mathbb{P}}^m)={\mathbb{Z}}^2$, generated freely by the pullbacks of coordinate hyperplanes. Restrict a divisor to the standard affine chart, where unique factorization makes it principal, and subtract this principal Weil divisor to leave only the two boundary hyperplanes. Any principal relation between them would be given by a rational function whose divisor vanishes on the chart; unique factorization then makes that function a constant, proving independence. The same argument gives $\mathrm{Cl}({\mathbb{P}}^r)=\mathbb{Z}$.