The orders of the elements in a cyclic group of order 12


Find the orders of each element of the additive group \mathbb{Z}/(12).
\overline{n}Reasoning|\overline{n}|
\overline{0}1
\overline{1}\overline{1} + \overline{1} = \overline{2}
\overline{2} + \overline{1} = \overline{3}
\overline{3} + \overline{1} = \overline{4}
\overline{4} + \overline{1} = \overline{5}
\overline{5} + \overline{1} = \overline{6}
\overline{6} + \overline{1} = \overline{7}
\overline{7} + \overline{1} = \overline{8}
\overline{8} + \overline{1} = \overline{9}
\overline{9} + \overline{1} = \overline{10}
\overline{10} + \overline{1} = \overline{11}
\overline{11} + \overline{1} = \overline{0}		12
\overline{2}\overline{2} + \overline{2} = \overline{4}
\overline{4} + \overline{2} = \overline{6}
\overline{6} + \overline{2} = \overline{8}
\overline{8} + \overline{2} = \overline{10}
\overline{10} + \overline{2} = \overline{0}		6
\overline{3}\overline{3} + \overline{3} = \overline{6}
\overline{6} + \overline{3} = \overline{9}
\overline{9} + \overline{3} = \overline{0}		4
\overline{4}\overline{4} + \overline{4} = \overline{8}
\overline{8} + \overline{4} = \overline{0}		3
\overline{5}\overline{5} + \overline{5} = \overline{10}
\overline{10} + \overline{5} = \overline{3}
\overline{3} + \overline{5} = \overline{8}
\overline{8} + \overline{5} = \overline{1}
\overline{1} + \overline{5} = \overline{6}
\overline{6} + \overline{5} = \overline{11}
\overline{11} + \overline{5} = \overline{4}
\overline{4} + \overline{5} = \overline{9}
\overline{9} + \overline{5} = \overline{2}
\overline{2} + \overline{5} = \overline{7}
\overline{7} + \overline{5} = \overline{0}		12
\overline{6}\overline{6} + \overline{6} = \overline{0}2
\overline{7}\overline{7} + \overline{7} = \overline{2}
\overline{2} + \overline{7} = \overline{9}
\overline{9} + \overline{7} = \overline{4}
\overline{4} + \overline{7} = \overline{11}
\overline{11} + \overline{7} = \overline{6}
\overline{6} + \overline{7} = \overline{1}
\overline{1} + \overline{7} = \overline{8}
\overline{8} + \overline{7} = \overline{3}
\overline{3} + \overline{7} = \overline{10}
\overline{10} + \overline{7} = \overline{5}
\overline{5} + \overline{7} = \overline{0}		12
\overline{8}\overline{8} + \overline{8} = \overline{4}
\overline{4} + \overline{8} = \overline{0}		3
\overline{9}\overline{9} + \overline{9} = \overline{6}
\overline{6} + \overline{9} = \overline{3}
\overline{3} + \overline{9} = \overline{0}		4
\overline{10}\overline{10} + \overline{10} = \overline{8}
\overline{8} + \overline{10} = \overline{6}
\overline{6} + \overline{10} = \overline{4}
\overline{4} + \overline{10} = \overline{2}
\overline{2} + \overline{10} = \overline{0}		6
\overline{11}\overline{11} + \overline{11} = \overline{10}
\overline{10} + \overline{11} = \overline{9}
\overline{9} + \overline{11} = \overline{8}
\overline{8} + \overline{11} = \overline{7}
\overline{7} + \overline{11} = \overline{6}
\overline{6} + \overline{11} = \overline{5}
\overline{5} + \overline{11} = \overline{4}
\overline{4} + \overline{11} = \overline{3}
\overline{3} + \overline{11} = \overline{2}
\overline{2} + \overline{11} = \overline{1}
\overline{1} + \overline{11} = \overline{0}		12







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