Algebra Solutions: Compute multiplicative orders in ZZ/(36)

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Compute multiplicative orders in ZZ/(36)

Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(36))^\times$ : $\overline{1}$ , $\overline{-1}$ , $\overline{5}$ , $\overline{13}$ , $\overline{-13}$ , and $\overline{17}$ .

$\overline{n}$	Reasoning	$\|\overline{n}\|$
$\overline{1}$		1
$\overline{-1}$	$\overline{-1} \cdot \overline{-1} = \overline{1}$	2
$\overline{5}$	The powers of $\overline{5}$ are $\overline{5}$ , $\overline{25}$ , $\overline{17}$ , $\overline{13}$ , $\overline{29}$ , and $\overline{1}$ .	6
$\overline{13}$	The powers of $\overline{13}$ are $\overline{13}$ , $\overline{25}$ , and $\overline{1}$ .	3
$\overline{-13}$	The powers of $\overline{-13}$ are $\overline{-13} = \overline{23}$ , $\overline{25}$ , $\overline{35}$ , $\overline{13}$ , $\overline{11}$ , and $\overline{1}$ .	6
$\overline{17}$	The powers of $\overline{17}$ are $\overline{1}$ .	2

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