Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine system
s(ab)= k, s(a)= ℓ, and s(b)= m
in odd integer variables a, b.When k =2 or k = 3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k =4, we give an upper bound for min{a, b} in terms of ℓ and m.