A factorial problem

\(d_n\) is the last nonzero digit of the decimal representation of \(n!\).

Is \(d_n\) eventually periodic?

Clarification: \(d_n\) is called "eventually periodic" if there exist \(k\) and \(n_0\) such that \(d_{n+k}=d_n\) for all \(n\geq n_0.\)

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