primitive root of modulo 9 and modulo 28

Question

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

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. — Herrpeter, Nov 6, 2014 at 5:26
There is no primitive root modulo $28$. For $9$, try everything, you will find them. (There are two.) — André Nicolas, 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. — Jyrki Lahtonen, Nov 6, 2014 at 5:28

Asinomás · Accepted Answer · 2014-11-06 05:37:41Z

2

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

$\lambda(28)=lcm(\varphi(4),\varphi(7))=lcm(2,6)=6$

answered Nov 6, 2014 at 5:30

Asinomás

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$\begingroup$ Note that $9$ has two primitive roots. $\endgroup$
– André Nicolas
Nov 6, 2014 at 5:34
$\begingroup$ thank you very much, and how can I find the order of modulo 9 and modulo 28? $\endgroup$
– Herrpeter
Nov 6, 2014 at 5:34
$\begingroup$ what do you mean? $\endgroup$
– Asinomás
Nov 6, 2014 at 5:41

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