**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 |