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Find a primitive root of (a) $U(\mathbb{Z}/121\mathbb{Z})$ and (b) $U(\mathbb{Z}/18\mathbb{Z})$.

Both of them are cyclic groups so primitive roots exist for both of them, but I don't know their generators. I know that $U(\mathbb{Z}/18\mathbb{Z}) \cong U(\mathbb{Z}/2\mathbb{Z})\times U(\mathbb{Z}/3^2\mathbb{Z})$. Also $\phi(18) = \phi(3^2) \phi(2) = 6$ and $\phi(121) = \phi(11^2) = 110$. So I need to find $a^{110} \equiv 1 \mod121$ and $b^{6} \equiv 1 \mod18$. I guess it will be easier to find $b$ by checking through units modular 18 but how do I find $a$?

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Hint: If $p$ is prime and $g$ is a primitive root mod $p$, then $g$ or $g+p$ is a primitive root mod $p^2$.

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  • $\begingroup$ If $p$ is prime and $g$ is a primitive root mod p, then $g$ or $g+p$ is a primitve root mod $p^2$. So is there a generalization for primitive root of $p^k$? $\endgroup$ – abuchay Mar 3 '18 at 15:23
  • $\begingroup$ @abuchay, yes, the same statement holds for $p^k$. $\endgroup$ – lhf Mar 3 '18 at 15:34
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    $\begingroup$ Thanks. Also after searching through internet, I found some other generalization here: people.math.gatech.edu/~mbaker/pdf/primroots.pdf $\endgroup$ – abuchay Mar 3 '18 at 15:51

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