Question

Find the order of the given element.

(a)$\mathbf{5}$in ${\mathit{U}}_{\mathbf{8}}$.

Short answer

The order of $5$in $U_8$is 2.

Step-by-step solution

Definition of groupUn

Let us denote the multiplicative group of units ${\mathbb{Z}}_n$in by ${\mathit{U}}_{\mathbf{n}}$.

Then, ${\mathit{U}}_{\mathbf{n}}$consists of all $a\in {\mathbb{Z}}_n$, such that $\mathbf{(}\mathbf{a}\mathbf{,}\mathbf{n}\mathbf{)}\mathbf{=}\mathbf{1}$(when a is considered as an ordinary integer).

Order of an element

Order of an element$\mathit{a}\mathbf{\in }\mathit{G}$ is defined as a positive integer n such tha${\mathit{a}}^{\mathbf{n}}\mathbf{=}\mathit{e}$, where $\mathit{e}$is the identity element in$\mathit{G}$ .

Order of 5 inU8U8

We have to find the order of 5 in $U_8$.

Here,${\left[5\right]}_8^2={\left[25\right]}_8={\left[1\right]}_8$

So, the order of $5$in $U_8$is 2.