I've heard all kinds of different ways to solve this problem, yet haven't been able to apply them specifically to the number 8 (Worked fine for 6 for example). I'd love to see a well-explained solution, is possible. Thank you.
2026-03-29 20:38:39.1774816719
How do I find all n values for which the equation $\phi (n) = 8$ holds?
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Hint: If $p^e$ divides $n$, then $\varphi(p^e)=p^{e-1}(p-1)$ divides $\varphi(n)$. Therefore:
$p=2$ and $e-1 \le 3$, or
$p$ is odd and $e=1$ and $p-1$ divides $8$.
This limits the possible candidates for $p^e$. You then need to argue how these candidates can be combined to yield $\varphi(n)=8$.