Let a series be defined as \[a_0=1\] \[a_n=\frac{n!}{a_{n-1}},n>0\].

Now consider the following statements.

\(a_n\) is an integer for \(n \ge 0\).

\(a_n\) is odd if and only if \(n\) is odd.

\(a_n\) is even, for positive \(n\), if and only if \(n\) is even

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