Question 179
For a finite non-empty set of primes $P$, let $|P|$ be the size of the set $P$, and $m(P)$ be 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 $\min(P) \gt |P|$;
(ii) Show that $m(P) \lt (|P| + 1)(2^{|P|} - 1)$.
(i) Show that $|P| \le m(P)$, with equality if and only if $\min(P) \gt |P|$;
(ii) Show that $m(P) \lt (|P| + 1)(2^{|P|} - 1)$.