An algebra problem

Algebra Level 3

Let \( S=\left\{ 1,2,3,4,5,6,7,8,9 \right\} \). Consider a function \(f:S\rightarrow S\) defined in the following table:

\[ \begin{array} { | c | c | } \hline x & f(x) \\ \hline 1 & 8 \\ 2 & 3 \\ 3 & 5 \\ 4 & 7 \\ 5 & 2 \\ 6 & 9 \\ 7 & 6 \\ 8 & 1 \\ 9 & 4 \\ \hline \end{array} \]

Then, the minimum integer value of \(n\) such that \( \underbrace { f(f(...(f } (x)))=x \) \(\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \)\(n\) times

for every \(x\in S\) is...


Source: 1st round of third level 2013 Brazilian Mathematics Olympiad(OBM).
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