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

Find the orders of each element of the additive group $\mathbb{Z}/(12)$.

Solution: For an element $n$ in $\mathbb{Z}/(12)$, the order of $\bar n$ is the smallest positive number $m$ such that $mn$ is a multiple of 12. Therefore, we have the following table.

element $\bar n$ order
$\bar 0$ 1
$\bar 6$ 2
$\bar 4$, $\bar 8$ 3
$\bar 3$, $\bar 6$, $\bar 9$ 4
$\bar 2$, $\bar 10$ 6
$\bar 1$, $\bar 5$, $\bar 7$, $\bar 11$ 12


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