Find all n such that:
$\displaystyle \varphi(n)=12$
The $\displaystyle \varphi(*)$ is a multiplicative function so that if m and n are coprime, then...
$\displaystyle \varphi(m\cdot n)= \varphi(m)\cdot \varphi(n)$ (1)
For $\displaystyle n=1$ and $\displaystyle n=2$ is $\displaystyle \varphi(n)=1$ and $\displaystyle \forall n>2$ $\displaystyle \varphi(n)$ is an even number, so that the solutions of the equation...
$\displaystyle \varphi (m\cdot n)=12$ (2)
... are obtained by systematic search of couples of coprime numbers m and n for which is...
$\displaystyle \varphi(m)=12, \varphi(n)=1$
$\displaystyle \varphi(m)=6, \varphi(n)=2$ (3)
Since $\displaystyle \varphi(13)=12$ solutions are...
$\displaystyle 13 \cdot 1 =13$
$\displaystyle 13\cdot 2 = 26$
Since $\displaystyle \varphi(7)=6$ also solutions are...
$\displaystyle 7 \cdot 3 = 21$
$\displaystyle 7 \cdot 4= 28$
$\displaystyle 7 \cdot 6 = 42$
Finally is $\displaystyle \varphi (9)= 6$ so that solution is...
$\displaystyle 9\cdot 4 = 36$
The numbers k for which is $\displaystyle \varphi(k)=12$ are...
$\displaystyle k=13,21,26,28,36,42$
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Hi, thank you for directing me here. Its good to see an example, but i dont understand this method.
Why do you multiply 7 by 3, 4 and 6? And how do you get $\displaystyle \phi(7) = 6$ and $\displaystyle \phi(9) = 6$. I know you can easily work these out but aren't both of these just an instance of the same sort of question as $\displaystyle \phi(m) = 12$? Aren't there more $\displaystyle \phi(m) = 6$?
Sorry if i'm being a bit slow, all this is very confusing for me.
The basic assumptions are...
a) is $\displaystyle \varphi(n)=1$ only for $\displaystyle n=1$ and $\displaystyle n=2$. For all $\displaystyle n>2$ , $\displaystyle \varphi (n)$ is an even number...
b) is $\displaystyle \varphi(m\cdot k)=\varphi(m)\cdot \varphi(k)$ if and only if m and k are coprime...
On the basis of b) the identification of an n for which $\displaystyle \varphi(n)=12$ is for the fact that must be $\displaystyle n=m\cdot k$ where m and k are coprime and $\displaystyle \varphi(m)\cdot \varphi(k) = 12$. There are the following possibilities...
$\displaystyle \varphi(m)=12$ , $\displaystyle \varphi(k)=1$
$\displaystyle \varphi(m)=6$ , $\displaystyle \varphi(k)=2$
The situation...
$\displaystyle \varphi(m)=4$ , $\displaystyle \varphi(k)=3$
... as proposed by tonio in the original post, is impossible on the basis of a)...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$