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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?

Herrpeter
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  • and how can I find the order of modulo 9 and modulo 28? Thank you very much for your help. – Herrpeter Nov 06 '14 at 05:26
  • See http://math.stackexchange.com/questions/584922/integer-m-has-primitive-root-if-and-only-if-the-only-solutions-of-the-congruen – lab bhattacharjee Nov 06 '14 at 05:28
  • There is no primitive root modulo $28$. For $9$, try everything, you will find them. (There are two.) – André Nicolas Nov 06 '14 at 05:28
  • 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 06 '14 at 05:28

1 Answers1

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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

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

Asinomás
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