Significance of a polynomial

Algebra Level 3

\[\large f(1)=f(2)=f(3) = \ldots = f(4535) = f(4536)=0\]

If \(f(x)\) is a monic polynomial of degree 4537, such that the equation above is satisfied and that \( f( 4537) - f(0) = N ! \) where \(N\) is a positive integer, what is \(N\)?

Bonus: Can you generalize this?

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