MathDB

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.

Problem Statement