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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?

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Why don't you include a link to your earlier question, where an answer was posted that used latin squares? –  Gerry Myerson Oct 29 '12 at 4:50
I will, sorry, I apologize. –  TTTY Oct 29 '12 at 4:53
I find your description hard to parse. Maybe you should include an example when $n=3$ or something. –  Greg Martin Oct 29 '12 at 7:29
Edited the question. Hope this is clear. –  TTTY Oct 29 '12 at 8:24

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