1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding Units Modular Arithmetic

  1. Jan 7, 2015 #1
    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. jcsd
  3. Jan 7, 2015 #2

    Mark44

    Staff: Mentor

    Please do not delete a post just because you have found a solution.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Finding Units Modular Arithmetic
  1. Modular arithmetic (Replies: 2)

Loading...