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So, I know that the integers that divide $16$ are $\{1, 2, 4, 8, 16\}$. And so the primes are $\{2, 3, 5, 9, 17\}$. Now I am stuck on how to get my possible $n$'s. I know the answers are $n = 17, 34, 60, 40, 48, 32$ or ($17, 2\cdot 17, 2^2\cdot 3\cdot 5, 2^3\cdot 5, 2^4\cdot 3, 2^5$). But how do I find this?

Thank you very much!

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As Quang Hoang points out, 9 is not a prime number, so the divisors of n are a subset $S \subset \{2,3,5,17\}$. If you know $S$, you can calculate $n$. For example with $S=\{2,3,5\}$ you get $\phi(n) = n\cdot\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{4}{5}=16 \Rightarrow n=60$. The equation delivers a valid solution iff its result is an integer and its prime factors are exactly the ones in $S$. So you need to try $2⁴=16$ subsets and you are done.

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