RMM 2010 Problem 1

For a finite non empty set of primes $P$ , let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$ .

(i) Show that $|P|\le m(P)$ , with equality if and only if $\min(P)>|P|$ .

(ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$ .

(The number $|P|$ is the size of set $P$ )

Dan Schwarz, Romania