Composite numbers dividing binomial coefficients

Let \(k\) be the number of composite integers \(n\) that satisfy the condition that \(n\) is a factor of \(\dbinom{n}{r}\) for all \(r\in \{ 1, 2, 3, \ldots, n-1\}.\) Which one of the following statements is always true for \(k\)? Prove that your answer is right.

Note: An integer number is called composite if it is greater than 1 and has at least a factor that is different from 1 and from itself.

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