Question 107
We say that a natural number is round if its number of divisors is a perfect square. For a round natural number $n$ with $d^{2}$ divisors we construct a $d \times d$ table and fill its cells with the divisors of $n$. At each step, we may choose a row of the table and move the divisor in column 1 to column 2, the one in column 2 to column $3, \ldots$, the one in column $d$ to column 1. Any configuration of the table that we reach is called feasible if there exists a column which has the elements $a_{1}, a_{2}, \ldots, a_{d}$ in this order and $a_{1}\mid a_{2}\mid \ldots \mid a_{d}$ or $a_{d}\mid a_{d-1}\mid \ldots \mid a_{1}$.
Determine the round numbers $n \geqslant 2$ for which, however the table of its divisors is initially filled, there exists a feasible configuration that can be reached after a finite number of steps.
Determine the round numbers $n \geqslant 2$ for which, however the table of its divisors is initially filled, there exists a feasible configuration that can be reached after a finite number of steps.