Let $a_1,a_2,\cdots,a_{2000}$ be distinct positive integers such that $1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000$ such that the LCM (least common multiple) of any two of them is $\geq 4000$. Show that $a_1 \geq 1334$
Problem
Source: SMO Senior 2019 Q3
Tags: number theory, least common multiple
29.06.2019 13:29
This is also a classical result :v In fact, the general statement hold: Let $a_1, a_2, a_3, \dots, a_n$ be distinct positive integers such that $1 \le a_1 < a_2 < \dots < 2n$ such that the LCM of any two of them is $\ge 2n$. Show that $a_1 \ge \left \lfloor \frac{2n}{3} \right \rfloor$.
29.06.2019 14:15
GorgonMathDota wrote: This is also a classical result :v In fact, the general statement hold: Let $a_1, a_2, a_3, \dots, a_n$ be distinct positive integers such that $1 \le a_1 < a_2 < \dots < 2n$ such that the LCM of any two of them is $\ge 2n$. Show that $a_1 \ge \left \lfloor \frac{2n}{3} \right \rfloor$. Can you show the source of this classical problem?
30.06.2019 18:59
We may consider the general case of $a_1, a_2, a_3, \dots, a_n$ be distinct positive integers such that $1 \le a_1 < a_2 < \dots < 2n$ such that the LCM of any two of them is $\ge 2n$.