Did you say "maximum"? Oh, come on!

Let \(A = \{1, 2, 3, \ldots, 20\}\) be the set of the first 20 positive integers. For each \(X\) that is a 15-element subset of \(A\), let \(p(X)\) be the product of its elements. What is the maximum integer \(n\) such that \(n\) divides \(p(X)\) for all choices of \(X\)?

From the Brazilian Maths Olympiads (OBM) 2013.
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