A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.
Source: Baltic Way 2000
Tags: algebra proposed, algebra
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.