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I don't know where to start with this problem so please help. The problem is:

Find all integers n such that $\phi(n) = 80$.

Martin Sleziak
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fittychap12
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    Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. Also, some would consider your post rude because it is a command ("Find"), not a request for help, so please consider rewriting it. – Zev Chonoles Mar 28 '13 at 06:08
  • You know a formula for $\varphi(n)$, right? That should be an obvious place to start, even if it's not obvious where to go from there. –  Mar 28 '13 at 07:46

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Hint: If the prime factorization of $n$ is $p_1^{a_1}\cdots p_r^{a_r}$, then $$\phi(n)=\phi(p_1^{a_1})\cdots\phi(p_r^{a_r})=p_1^{a_1-1}(p_1-1)p_2^{a_2-1}(p_2-1)\cdots p_r^{a_r}(p_r-1).$$ The prime factorization of $80$ is $2^4\cdot 5$. What are the possible ways the $2$'s and the $5$ can be distributed among the factors on the right side of this equation?

Zev Chonoles
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