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Compute multiplicative orders in Z/(n)


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.12

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


Solution:

$\overline{n}$       Reasoning       Order
$\overline{1}$1
$\overline{-1}$$\overline{-1} \cdot \overline{-1} = \overline{1}$2
$\overline{5}$$\overline{5} \cdot \overline{5} = \overline{25} = \overline{1}$2
$\overline{7}$$\overline{7} \cdot \overline{7} = \overline{49} = \overline{1}$2
$\overline{-7}$$\overline{-7} = \overline{5}$2
$\overline{13}$$\overline{13} = \overline{1}$1


Linearity

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