Find the largest integer \(n\), such that for some non-constant cubic polynomial \(f(x)\) with integer coefficients,
\[ f(1) \mid f(2), f(2) \mid f(3), \ldots f(n-1) \mid f(n). \]
Details and assumptions
For (possibly negative or zero) integers \(m\) and \(k\) the notation \(m|k\) means that \(k=m\cdot i\) for some integer \(i.\)
The divisibility condition is a statement about integers, not polynomial.