2013 OIM Problems/Problem 1

Problem

A set $S$ of positive integers is called "canalero" if for any three numbers $a, b, c \in S$ , all different, it holds that $a$ divides $bc$ , $b$ divides $ca$ , and $c$ divides $ab$ .

1. Show that, for any finite set of positive integers $\left\{c_1, c_2, \cdots , c_n\right\}$ , there exist infinite positive integers $k$ such that the set $\left\{kc_1, kc_2, \cdots , kc_n\right\}$ is a canalero.

2. Show that, for any integer $n \ge 3$ , there is a canalero set that has exactly $n$ elements and no integer greater than 1 divides all its elements.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2013 OIM Problems/Problem 1

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