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How can I show that $(\mathbb Z /n\mathbb Z)^*$ is cyclic or not for a given $n$?

If $n$ is small like 10 or 11 then we can compute the number of elements and then can find an element whose order is $\phi(n)$. But if $n$ is large enough then how can I say it is cyclic or not?

Here $(\mathbb Z /n\mathbb Z)^*$ denotes the set of all units in $\mathbb Z/n\mathbb Z$.

Need some help please..

Mini_me
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  • https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n – Tsemo Aristide Jun 17 '17 at 11:28
  • Then according to wiki $(\mathbb Z/ 8\mathbb Z)^$ is isomorphic to $C_2 \times C_2$ and the later is non-cyclic . But in $(\mathbb Z/ 8\mathbb Z)^$ there is an element 5 (mod 8) $s.t.$ $5^4=1(mod 8)$ and this implies that $(\mathbb Z/ 8\mathbb Z)^*$ is cyclic. Am I wrong somewhere? Please check @TsemoAristide – Mini_me Jun 17 '17 at 11:49
  • Yes I see it now @SahibaArora. Should I delete this post? – Mini_me Jun 17 '17 at 11:52
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    The order of $5$ in the group is not $4$, since $5^2=1$ – Exit path Jun 17 '17 at 11:52
  • ooh yeah..I got it @leibnewtz – Mini_me Jun 17 '17 at 11:53

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