Finding Units Modular Arithmetic

  • Thread starter auru
  • Start date
  • #1
10
0

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## ∈ ℤn 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

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.
 

Answers and Replies

  • #2
34,688
6,394
Please do not delete a post just because you have found a solution.
 

Related Threads on Finding Units Modular Arithmetic

  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
2K
Replies
2
Views
3K
Replies
7
Views
1K
Replies
5
Views
10K
  • Last Post
Replies
4
Views
667
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
1
Views
473
Top