<s>[INST] primitive residue classes modulo 32 $\mathbb Z_{32}^*$ is the primitive residue classes modulo 32. How is it possible to show that $\mathbb Z_{32}^*$ is generated by 5 and -1, without showing it for every element of $\mathbb Z_{32}^*$=$\{1,3,5,7,9,11,...,31\}$
 [/INST] Let $n > 1$ be an integer.
We consider the group $G = (\mathbb{Z}/2^n\mathbb{Z})^*$.
Clearly $|G| = 2^{n-1}$.
Let $e \ge 2$ be an integer.
Let $k$ be an odd integer.
$(1 + 2^e k)^2 = 1 + 2^{e+1}k + 2^{2e} k^2 = 1 + 2^{e+1}(k + 2^{e-1}k^2)$.
Note that $k + 2^{e-1}k^2$ is odd.
Hence, by induction on $m \ge 1$, $(1 + 2^e k)^{2^m} = 1 + 2^{e+m}k'$ for some odd integer $k'$.
Since $5 = 1 + 2^2$, $5^{2^m} = 1 + 2^{m + 2}k'$, where $k'$ is odd.
Hence the order of $5$ in $G$ is $2^{n-2}$.
Let $H$ be the subgroup of $G$ generated by $5$.
Suppose $-1 \in H$.
There exists integer $m$ such that $-1 \equiv 5^m$ (mod $2^n$).
Since $5 \equiv 1$ mod $4$, $-1 \equiv 1$ (mod $4$).
This is a contradiction.
Hence $G$ is generated by $-1$ and $5$.
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