Finite cyclic group of residue set of integers

Question

Recall that $Z_n$ is the finite cyclic group with n elements and group multiplication given by $x * y :=(x+y) \pmod n$. For $n = 5$, find all $x \in Z_n$ such that $ \langle x \rangle=Z_n$

So for this, do I just enumerate the $x$'s such that their multiples encompass all of $Z_5$. Such that:

$\langle 1 \rangle=\{0,1,2,3,4\}=Z_5$

$\langle 2 \rangle=\{0,2,4,1,3\}=Z_5$

$\langle 3 \rangle=\{0,3,1,4,2\}=Z_5$

$\langle 4 \rangle=\{0,4,3,2,1\}=Z_5$

So, in this case all of the $x$ that are equal to $Z_5$ are $x=1,2,3,4$?

Answer 1

Your working is correct. In general $\langle x\rangle=Z_n$ if and only if $\mathrm{gcd}(x,n)=1$.

Answer 2

The generators of $Z_n$ are precisely all those elements less than and co-prime to $n$. For example in your case n=5 Just count the numbers which are less than 5 and co-prime to n. There will be $\phi (n)$ elements in total where $$\phi (n)= n (1-(1/p_1))(1-(1/p_2))..... (1-(1/p_n))$$ Where $n=p_1p_2.....p_n$