Let \(f(x) = \left\{ \begin{array}{cl} \frac{x}{21} & \text{ if }x\text{ is a multiple of 3 and 7}, \\ 3x & \text{ if }x\text{ is only a multiple of 7}, \\ 7x & \text{ if }x\text{ is only a multiple of 3}, \\ x+3 & \text{ if }x\text{ is not a multiple of 3 or 7}. \end{array} \right.\)If \(f^a(x)\) means the function is nested \(a\) times (for example, \(f^2(x)=f(f(x))\)), what is the

smallest value of \(a\) greater than 1 that satisfies \(f(2)=f^a(2)\)?

supremecheetah May 22, 2023