BxMO 2017 Problem 4

A Benelux n-square (with $n\geq 2$ ) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes.

(a) Prove that, in each Benelux n-square (with $n \geq 2$ ), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square minimal if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.