  We give a relatively short proof of one of the central cases of the main
theorem from the paper "The distribution of integers with a divisor in a given
interval", math.NT/0401223. Namely, we determine the order of magnitude of the
number of integers <=x with a divisor in (y,2y]. The lower bound uses a
different argument than that in the aforementioned paper. As a corollary, we
deduce the order of magnitude for the number of distinct products in an N x N
multiplication table.
