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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 \(f(2)=f^a(2)\)?

 May 22, 2023
 #1
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We have that f(2) = 2/21 = 1/14. Since 2 is not a multiple of 3 or 7, we have that f^a(2) = 2 + 3 = 5. Therefore, we need to find the smallest value of a such that 1/14 = 5. This is never true, so the smallest value of a is 2​.

 May 22, 2023
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Incorrect, the answer is 7.

 May 22, 2023

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