Exercise 18, § 1

Let M be the multiplicative monoid generated by a finite number of integers $q_i > 1$ ($1 \leq i \leq m$). Show that, for each integer $k > 0$, there exist k consecutive integers which do not belong to M (if $h_1, \ldots, h_m$ are m integers > 1, and k is the greatest number of consecutive integers none of which is divisible by any of the $h_i$, then $(k + 1)^1 \leq h_1^{-1} + \ldots + h_m^{-1}$; now notice that, for all $r > 0$, every sufficiently large number in M is divisible by at least one $q_i^+$). Obtain a new proof of Prop. 5 of VII, p. 5 from this result.