Finding Units Modular Arithmetic

[auru]

Homework Statement

I am required to find the units of ℤ₈.

Homework Equations

I have that
$\bar{a}$ = [a]_n = { a + kn, k ∈ ℤ }
$u$ ∈ ℤ_n is a unit if $u$ divides $\bar{1}$.

The Attempt at a Solution

I'm not sure how to go about this. My lecturer wrote out the multiplication table for ℤ₈ and simply noted that by inspection of the table, the units are: $\bar{1}$, $\bar{3}$, $\bar{5}$, $\bar{7}$.

So I have the multiplication table

ℤ₈ $\bar{0}$, $\bar{1}$, $\bar{2}$, $\bar{3}$, $\bar{4}$, $\bar{5}$, $\bar{6}$, $\bar{7}$,
$\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$,
$\bar{1}$, $\bar{0}$, $\bar{1}$, $\bar{2}$, $\bar{3}$, $\bar{4}$, $\bar{5}$, $\bar{6}$, $\bar{7}$,
$\bar{2}$, $\bar{0}$, $\bar{2}$, $\bar{4}$, $\bar{6}$, $\bar{0}$, $\bar{2}$, $\bar{4}$, $\bar{6}$,
$\bar{3}$, $\bar{0}$, $\bar{3}$, $\bar{6}$, $\bar{1}$, $\bar{4}$, $\bar{7}$, $\bar{2}$, $\bar{5}$,
$\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$,
$\bar{5}$, $\bar{0}$, $\bar{5}$, $\bar{2}$, $\bar{7}$, $\bar{4}$, $\bar{1}$, $\bar{6}$, $\bar{3}$,
$\bar{6}$, $\bar{0}$, $\bar{6}$, $\bar{4}$, $\bar{2}$, $\bar{0}$, $\bar{6}$, $\bar{4}$, $\bar{2}$,
$\bar{7}$, $\bar{0}$, $\bar{7}$, $\bar{6}$, $\bar{5}$, $\bar{4}$, $\bar{3}$, $\bar{2}$, $\bar{1}$,

By inspection of the table, I see that $\bar{1}$, $\bar{3}$, $\bar{5}$, $\bar{7}$ all yield rows where each product is unique, indicating that they are the units of ℤ₈. However, I'd like to know of a more concrete way of finding the units of ℤ_n, if that is possible.

 

[Mark44]

Please do not delete a post just because you have found a solution.