1

Find all 10 numbers n such that $φ(n) = 24$. Show that there are no others.

I'm so lost. I know that I have to find $φ(n) = 2,3,4,6,8,12$. But don't really know where I'm going from there.

user35603
  • 3,002
user275617
  • 11
  • 1
    Hint (referring to the title question): factor $n=\prod,p_i^{a_i}$, so that $\varphi(n)=\prod (p_i-1)p_i^{a_i-1}$. Can you first figure out which prime $p_i$ are possible factors of $n$? – lulu Sep 30 '15 at 19:24
  • 1
    The good news is that you don't have to find all numbers $n$ for which $\varphi(n)=2,3,4,6,8,12$, because it's not going to be of any use anyway. – Ivan Neretin Sep 30 '15 at 19:24
  • See also http://math.stackexchange.com/questions/23947/how-to-solve-the-equation-phin-k, http://math.stackexchange.com/questions/284003/computing-n-such-that-phin-m. – lhf Sep 30 '15 at 19:42
  • @IvanNeretin, it certainly is useful. – lhf Sep 30 '15 at 19:43
  • See also http://www.jstor.org/stable/2308462. – lhf Sep 30 '15 at 19:44

1 Answers1

3

Hint: $\phi(n)$ is a product of $p^k(p-1)$ for all primes $p$ dividing $n$. Which $p^k$ and $p-1$ divide $24=2^3 \cdot 3$?

Start with this: write $n=2^e m$, with $m$ odd. Then $\phi(n)=\phi(2^e)\phi(m)$ and you can bound $e$.

lhf
  • 216,483