**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.14**

Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(36))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{13}$, $\overline{-13}$, and $\overline{17}$.

Solution:

$\overline{n}$ | Reasoning | order |
---|---|---|

$\overline{1}$ | 1 | |

$\overline{-1}$ | $\overline{-1} \cdot \overline{-1} = \overline{1}$ | 2 |

$\overline{5}$ | The powers of $\overline{5}$ are $\overline{5}$, $\overline{25}$, $\overline{17}$, $\overline{13}$, $\overline{29}$, and $\overline{1}$. | 6 |

$\overline{13}$ | The powers of $\overline{13}$ are $\overline{13}$, $\overline{25}$, and $\overline{1}$. | 3 |

$\overline{-13}$ | The powers of $\overline{-13}$ are $\overline{-13} = \overline{23}$, $\overline{25}$, $\overline{35}$, $\overline{13}$, $\overline{11}$, and $\overline{1}$. | 6 |

$\overline{17}$ | The square of $\overline{17}$ is $\overline{1}$. | 2 |