Binomial Congruence

Consider the following binomial coefficient: \[\dbinom{n}{\big\lceil\frac{n}{2}\big\rceil},\] where \(\lceil \cdot \rceil\) is the ceiling function.

If it is divisible by \(n\), what can be said about \(n \geq 2\)?

(1) Only prime odd integers work.
(2) None of the even composite integers work.
(3) All odd integers work.

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