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Determine the solutions of $\varphi (n) = 30$.

I already understood that the possible solutions will be $31$ because it is prime and $62$ because it is double $31$, but I don't know how to justify that there is no more possible solution and prove that these two are the only possible ones.

lulu
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Manuel Rodrigues
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  • Well, do you know how to relate $\varphi(n)$ to the prime factorization of $n$? – lulu Nov 11 '22 at 15:45
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    HINT: $30$ contains a factor of $5$. How can that arise? Either from $n$ containing a power of $5$, or from $n$ containing a prime factor which is $1$ greater than a multiple of $5$. See if you can use those facts to limit what $n$ can give rise to $\phi(n)=30$ – Keith Backman Nov 11 '22 at 15:53
  • I already know that ϕ(n) is based on the prime divisors of a given number, but I don't know how to exclude the divisors of 30 that will not be needed to calculate aϕ(n)= 30 for example (2,3,5) I don't know if I have to go there or somewhere else – Manuel Rodrigues Nov 11 '22 at 16:02
  • How to solve the equation $\phi(n) = k$? might be useful – Sil Nov 11 '22 at 18:56
  • @ManuelRodrigues If you still haven't seen the answer from the hints, reply to this comment and I will post an answer. – Keith Backman Nov 12 '22 at 14:26

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