# Homework Help: Finding Units Modular Arithmetic

1. Jan 7, 2015

### auru

1. The problem statement, all variables and given/known data

I am required to find the units of ℤ8.

2. Relevant equations

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

3. 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

8 $\bar{0}$, $\bar{1}$, $\bar{2}$, $\bar{3}$, $\bar{4}$, $\bar{5}$, $\bar{6}$, $\bar{7}$,
$\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$, $\bar{0}$,
$\bar{1}$, $\bar{0}$, $\bar{1}$, $\bar{2}$, $\bar{3}$, $\bar{4}$, $\bar{5}$, $\bar{6}$, $\bar{7}$,
$\bar{2}$, $\bar{0}$, $\bar{2}$, $\bar{4}$, $\bar{6}$, $\bar{0}$, $\bar{2}$, $\bar{4}$, $\bar{6}$,
$\bar{3}$, $\bar{0}$, $\bar{3}$, $\bar{6}$, $\bar{1}$, $\bar{4}$, $\bar{7}$, $\bar{2}$, $\bar{5}$,
$\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$, $\bar{0}$, $\bar{4}$,
$\bar{5}$, $\bar{0}$, $\bar{5}$, $\bar{2}$, $\bar{7}$, $\bar{4}$, $\bar{1}$, $\bar{6}$, $\bar{3}$,
$\bar{6}$, $\bar{0}$, $\bar{6}$, $\bar{4}$, $\bar{2}$, $\bar{0}$, $\bar{6}$, $\bar{4}$, $\bar{2}$,
$\bar{7}$, $\bar{0}$, $\bar{7}$, $\bar{6}$, $\bar{5}$, $\bar{4}$, $\bar{3}$, $\bar{2}$, $\bar{1}$,

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 ℤn, if that is possible.

2. Jan 7, 2015

### Staff: Mentor

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

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