Euler Totient Function Equations (Problem $3$ )

$\phi_n$ denotes how many times Euler's totient function can be applied before the result reaches $0$ - an iterative form of Euler's totient function.

From this:

$\phi(180) = 48$ $(1)$

$\phi(48) = 16$ $(2)$

$\phi(16) = 8$ $(3)$

$\phi(8) = 4$ $(4)$

$\phi(4) = 2$ $(5)$

$\phi(2) = 1$ $(6)$

Therefore, $n = \fbox 6$