Is there a set of $n$ sets that has all the following:
1) Inside each set, there are $n$ different numbers. We can form a product of $n$ numbers from some sets. If one matrix is multiplied more than once when multiplying matrices to get the product, this becomes some multiples of $x$. Otherwise, if the product is not the multiples of $x$.
Will mutually orthogonal latin square methods help? (Link: Finding the greatest common factor that matches some constraints)
Does such $n$ exist for all natural numbers bigger than 4?