Exercise 1.1.13 (Additive orders of elements of $\mathbb{Z}/36 \mathbb{Z}$)

Question

Find the orders of the following elements of the additive group $\mathbb{Z}/36 \mathbb{Z}$: $\overline{1}, \overline{2}, \overline{6}, \overline{9}, \overline{10}, \overline{12}, \overline{-1}, \overline{-10}, \overline{-18}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
As in Exercise 11,
$$|\overline{k}| = \frac{36}{k \wedge 36}.$$
This gives
\begin{center}
\begin{tabular}{c|ccccccccc}
$x$ & $\ \overline{1}$ & $\ \overline{2}$ & $\ \overline{6}$ & $\ \overline{9}$ &$\ \overline{10}$ &$\ \overline{12}$ &$\ \overline{-1}$ &$\ \overline{-10}$ &$\ \overline{-18}$\
\hline
$|x|$  & 36 & 18 & 6 & 4 & 18 &3 & 36 & 18 & 2
\end{tabular}
\end{center}

\end{proof}
With Sagemath:
\begin{verbatim}
sage: G = Integers(36)
sage: for k in [1,2,6,9,10,12,-1,-10,-18]:
....:     print(k,G(k).additive_order())
....:     
(1, 36)
(2, 18)
(6, 6)
(9, 4)
(10, 18)
(12, 3)
(-1, 36)
(-10, 18)
(-18, 2)
\end{verbatim}

$x$	$\bar{1}$	$\bar{2}$	$\bar{6}$	$\bar{9}$	$\bar{10}$	$\bar{12}$	$\bar{- 1}$	$\bar{- 10}$	$\bar{- 18}$

$\| x \|$	36	18	6	4	18	3	36	18	2

Exercise 1.1.13 (Additive orders of elements of $\mathbb{Z}/36 \mathbb{Z}$)

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Exercise 1.1.13 (Additive orders of elements of $\mathbb{Z}/36 \mathbb{Z}$)

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