A [i]Benelux n-square[/i] (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 [i]minimal[/i] 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.
### Part (a)

We need to prove that in each Benelux \( n \)-square (with \( n \geq 2 \)), there exists a cell containing a number which is at least \( 2n^2 \).

1. **Distinct Positive Integers**: Since the \( n^2 \) positive integers in the grid are pairwise distinct, the smallest possible set of these integers is \( \{1, 2, 3, \ldots, n^2\} \).

2. **Maximum Value**: The maximum value in this set is \( n^2 \).

3. **GCD Conditions**: For each row and each column, we compute the greatest common divisor (GCD) of the \( n \) numbers in that row/column, and we obtain \( 2n \) different outcomes.

4. **Minimum GCD Values**: The smallest possible distinct GCD values for \( 2n \) different outcomes are \( 1, 2, 3, \ldots, 2n \).

5. **Number of Distinct GCDs**: Since there are \( 2n \) distinct GCDs, and each GCD must be a divisor of some number in the grid, the numbers in the grid must be large enough to accommodate these GCDs.

6. **Conclusion**: If all numbers in the grid were less than \( 2n^2 \), it would be impossible to have \( 2n \) distinct GCDs. Therefore, there must be at least one number in the grid that is at least \( 2n^2 \).

\(\blacksquare\)

### Part (b)

We need to determine all \( n \geq 2 \) for which there exists a minimal Benelux \( n \)-square, where all \( n^2 \) numbers in the cells are at most \( 2n^2 \).

1. **Minimal Benelux \( n \)-square**: A minimal Benelux \( n \)-square is defined such that all \( n^2 \) numbers in the cells are at most \( 2n^2 \).

2. **GCD Constraints**: The GCDs of the rows and columns must be distinct and within the range \( 1, 2, \ldots, 2n \).

3. **Construction for \( n = 2 \)**:
   - Consider the grid:
     \[
     \begin{array}{cc}
     4 & 2 \\
     3 & 1 \\
     \end{array}
     \]
   - The GCDs of the rows are \( \gcd(4, 2) = 2 \) and \( \gcd(3, 1) = 1 \).
   - The GCDs of the columns are \( \gcd(4, 3) = 1 \) and \( \gcd(2, 1) = 1 \).
   - This grid satisfies the conditions for a minimal Benelux \( 2 \)-square.

4. **Construction for \( n = 4 \)**:
   - Consider the grid:
     \[
     \begin{array}{cccc}
     8 & 4 & 2 & 1 \\
     12 & 6 & 3 & 9 \\
     16 & 8 & 4 & 2 \\
     20 & 10 & 5 & 15 \\
     \end{array}
     \]
   - The GCDs of the rows and columns are distinct and within the range \( 1, 2, \ldots, 8 \).
   - This grid satisfies the conditions for a minimal Benelux \( 4 \)-square.

5. **General Case for Higher \( n \)**:
   - For \( n > 4 \), it becomes increasingly difficult to construct a grid where all numbers are at most \( 2n^2 \) while maintaining \( 2n \) distinct GCDs.
   - The constraints on the GCDs and the distinctness of the numbers make it impossible to construct such a grid for \( n > 4 \).

6. **Conclusion**: The only values of \( n \) for which there exists a minimal Benelux \( n \)-square are \( n = 2 \) and \( n = 4 \).

\(\blacksquare\)

The final answer is \( \boxed{ n = 2 } \) and \( n = 4 \).