Exercise 1.1.12 (Multiplicative order of the elements of $(\mathbb{Z}/12\mathbb{Z})^ imes$)

Question

Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/12\mathbb{Z})^	imes$: $\overline{1}, \overline{-1}, \overline{5}, \overline{7}, \overline{-7}, \overline{13}$.

Richard Ganaye · Accepted Answer

\begin{proof} For all $k \in \Z$,
$$\overline{k}  \in (\mathbb{Z}/12\mathbb{Z})^	imes \iff k \wedge n = 1\,$$
so
$$(\mathbb{Z}/12\mathbb{Z})^	imes = \{\overline{1}, \overline{-1}, \overline{5}, \overline{-5}\},$$
where all the elements have order $1$ (for $\overline{1}$) or $2$ (for $\overline{-1}, \overline{5}, \overline{-5}$). Since $\overline{13} = \overline{1}$, and $\overline{7} = \overline{-5}$, $\overline{-7} = \overline{5}$, we obtain
\begin{align*}
&|\overline{1}| = |\overline{13}| = 1,\
&|\overline{-1}| = |\overline{5}| = |\overline{7}| = |\overline{-7}| = 2.
\end{align*}
\end{proof}

Exercise 1.1.12 (Multiplicative order of the elements of $(\mathbb{Z}/12\mathbb{Z})^\times$)

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Exercise 1.1.12 (Multiplicative order of the elements of $(\mathbb{Z}/12\mathbb{Z})^\times$)

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