[math-fun] Zero matrix products question

Given N in Z+, what is the largest possible size f(N) of a set X of NxN matrices over Z such that 1) Any pair of them multiply to the zero matrix; 2) Each member of X has no common factor among all N^2 of its entries. ??? Having spent only a few minutes on this, it seems clear that f(N) >= 1 + floor(N/2)^2 (exercise). Maybe it's obvious, but I don't even see why f(N) must be finite (though I'd guess it is). One could also ask the question in a variety of other ways, such as for matrices whose entries are all nonnegative integers. Or over certain rings. --Dan