34 pages. Added reference [10] to a paper of Nair and Tenenbaum which contains a result similar to Lemma 2.2 and which appeared prior to the publication of this paper. Simplified the proof of Lemma 2.2 using ideas from [10]. Removed part (a) of Lemma 2.2, as it is now redundant. Version 5 remains the published version

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i