# primitive root of modulo 9 and modulo 28

What are the primitive roots of modulo 9 and modulo 28?

I tried to solve this problem after http://en.wikipedia.org/wiki/Primitive_root_modulo_n

Elementary example, but it seems to be to long for me. are there any opportunietes to give the primitive roots of modulo 9 and modulo 28?

• and how can I find the order of modulo 9 and modulo 28? Thank you very much for your help. Nov 6, 2014 at 5:26
• Nov 6, 2014 at 5:28
• There is no primitive root modulo $28$. For $9$, try everything, you will find them. (There are two.) Nov 6, 2014 at 5:28
• A primitive root modulo $n$ exists only if $n$ is a power of an odd prime, twice the power of an odd prime, or a factor of four. 9 fits the first group (2 is a primitive root), but 28 does not. Nov 6, 2014 at 5:28

One of the primitive root modulo $9$ is $2$ since we have $2,4,8,7,5,1$.
The number of primitive roots mod $n$ in case any exist is $\varphi(\varphi(n))$
To see why $28$ has no primitive roots us charmichaels theorem to see
$\lambda(28)=lcm(\varphi(4),\varphi(7))=lcm(2,6)=6$
• Note that $9$ has two primitive roots. Nov 6, 2014 at 5:34