Finding Units Modular Arithmetic

In summary, the task is to find the units of ℤ8 and the method used is by inspecting the multiplication table for ℤ8. The units are determined to be ##\bar{1}##, ##\bar{3}##, ##\bar{5}##, ##\bar{7}## based on the fact that they yield unique products in each row. However, the poster is interested in a more concrete method for finding the units of ℤn.
  • #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.
 
Physics news on Phys.org
  • #2
Please do not delete a post just because you have found a solution.
 

1. What is modular arithmetic?

Modular arithmetic is a mathematical system that deals with integers and their remainders when divided by a chosen number, also known as the modulus. It is a way to perform calculations on cyclic groups and is commonly used in computer science and cryptography.

2. How is modular arithmetic used in real life?

Modular arithmetic has many practical applications, such as in clock arithmetic, where the numbers on a clock repeat every 12 hours. It is also used in calculating interest rates, determining leap years, and in coding and encryption for secure communication.

3. What is the purpose of finding units in modular arithmetic?

The units in modular arithmetic refer to the numbers that have a multiplicative inverse within a given modulus. Finding these units is important in determining the solvability of equations and for finding solutions to problems in modular arithmetic.

4. How do you find units in modular arithmetic?

To find the units in modular arithmetic, you can use the Euclidean algorithm to find the greatest common divisor between the chosen modulus and each integer. If the greatest common divisor is 1, then the integer is a unit. Alternatively, you can use a table of remainders to determine which numbers have a multiplicative inverse within the given modulus.

5. Can units in modular arithmetic be negative?

Yes, units in modular arithmetic can be negative. The negative units refer to integers that have a multiplicative inverse within the modulus, but when multiplied by a positive integer result in a negative remainder. For example, in mod 5 arithmetic, -3 is a unit because -3 * 2 = -1, which has a remainder of 4 when divided by 5.

Suggested for: Finding Units Modular Arithmetic

Back
Top