  We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb
G_m^n$ to more general settings. More specifically, we analyze the height of a
closed subscheme of codimension at least $2$ inside an $n$-dimensional
Cohen-Macaulay projective variety, and show that this height is small when
evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of
$n+1$ effective divisors which are all numerically equivalent to some multiples
of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate
as an application of Vojta's conjecture, which is substituted by a more general
version of Schmidt's subspace theorem of Ru--Vojta in our proof.
