Algebra Solutions: Compute additive orders in ZZ/(n)

Compute additive orders in ZZ/(n)

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

$\overline{n}$	Reasoning	$\|\overline{n}\|$
$\overline{1}$	36 is the smallest multiple of 1 that is congruent to 0 mod 36.	36
$\overline{2}$	36 is the smallest multiple of 2 that is congruent to 0 mod 36.	18
$\overline{6}$	Multiples of $\overline{6}$ are $\overline{6}$ , $\overline{12}$ , $\overline{18}$ , $\overline{24}$ , $\overline{30}$ , $\overline{36} = \overline{0}$	6
$\overline{9}$	Multiples of $\overline{9}$ are $\overline{9}$ , $\overline{18}$ , $\overline{27}$ , $\overline{36} = \overline{0}$	4
$\overline{10}$	Multiples of $\overline{10}$ are $\overline{10}$ , $\overline{20}$ , $\overline{30}$ , $\overline{4}$ , $\overline{14}$ , $\overline{24}$ , $\overline{34}$ , $\overline{8}$ , $\overline{18}$ , $\overline{28}$ , $\overline{2}$ , $\overline{12}$ , $\overline{22}$ , $\overline{32}$ , $\overline{6}$ , $\overline{16}$ , $\overline{26}$ , $\overline{36} = \overline{0}$	18
$\overline{12}$	Multiples of $\overline{12}$ are $\overline{12}$ , $\overline{24}$ , $\overline{36} = \overline{0}$	3
$\overline{-1}$	36 is the smallest multiple of -1 that is congruent to 0 mod 36.	36
$\overline{-10}$	Multiples of $\overline{-10}$ are $\overline{-10} = \overline{26}$ , $\overline{16}$ , $\overline{6}$ , $\overline{32}$ , $\overline{22}$ , $\overline{12}$ , $\overline{2}$ , $\overline{28}$ , $\overline{18}$ , $\overline{8}$ , $\overline{34}$ , $\overline{24}$ , $\overline{14}$ , $\overline{4}$ , $\overline{30}$ , $\overline{20}$ , $\overline{10}$ , $\overline{0}$	18
$\overline{-18}$	$\overline{-18} + \overline{-18} = \overline{0}$	2

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