TITLE: Show that among any consecutive $16$ natural numbers one is coprime to all others
QUESTION [10 upvotes]: Show that among any consecutive $16$ natural numbers one is coprime to all others.
Is it useful to use the division algorithm on $16$?
$16k,16k+1,16k+2,...16k+15$

REPLY [3 votes]: Theorem. (S. S. Pillai) For $m\in\mathbb N$ the following statements are equivalent:
(1) there is a set of $m$ consecutive integers, none of which is relatively prime to all the others;
(2) $m\ge17$.
The smallest example of a set as in (1) is $\{2184,2185,\dots,2200\}$.
S. S. Pillai, On $m$ consecutive integers—III, Proc. Indian Acad. Sci. 13 (1941), 530–533.
Irene Gassko, Stapled sequences and stapling coverings of natural numbers, Electronic Journal of Combinatorics, Volume 3, Issue 1, 1996.