+1306 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 \(f(2)=f^a(2)\)?
We have that \begin{align*} f(2) &= \frac{2}{21} \ f^2(2) &= f(f(2)) = f \left( \frac{2}{21} \right) \ &= 3 \cdot \frac{2}{21} = \frac{2}{7} \ f^3(2) &= f(f^2(2)) = f \left( \frac{2}{7} \right) \ &= 7 \cdot \frac{2}{7} = 2 \ f^4(2) &= f(f^3(2)) = f(2) = \frac{2}{21}. \end{align*}Since f4(2)=f(2), the smallest value of a greater than 1 that satisfies f(2)=fa(2) is 4.
