**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.11**

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 |