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## Compute multiplicative orders in Z/(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 $\overline{17}$.

Solution:

$\overline{n}$Reasoningorder
$\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 square of $\overline{17}$ is $\overline{1}$.2

#### Linearity

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