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Can you please show me how to write Z12?  From the image, how did we get [8]12 = [20]12 .... rest is in the image. From where did the numbers come?    2. In Z7 , can you please show me how to get the multiplicative inverse of Z7 ?  How did the book list [1]7-1 = [1]7 = [2]7-1 = [4]7

Can you please show me how to write Z12?  From the image, how did we get [8]12 = [20]12 .... rest is in the image. From where did the numbers come?    2. In Z7 , can you please show me how to get the multiplicative inverse of Z7 ?  How did the book list [1]7-1 = [1]7 = [2]7-1 = [4]7

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 4TFE
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1.

Can you please show me how to write Z12? 

From the image, how did we get [8]12 = [20]12 .... rest is in the image. From where did the numbers come? 

 

2. In Z7 , can you please show me how to get the multiplicative inverse of Z7 ?  How did the book list [1]7-1 = [1]7 = [2]7-1 = [4]7 .... the rest is in the image. 

 

please do easily explain them to me so I can follow up with your steps and understand. Thank you. 

In Z7, each nonzero congruence class contains representatives which are rela- tively prime to 7, and so each nonzero congruence class has a multiplicative inverse. We can list them as [1]7' = [1]7, [2]7' = [4]7, [3]7' We did not need to list [4],' and [5]7' since, in general, if [a],' = [b]n, then [b],' = [a]n.
Transcribed Image Text:In Z7, each nonzero congruence class contains representatives which are rela- tively prime to 7, and so each nonzero congruence class has a multiplicative inverse. We can list them as [1]7' = [1]7, [2]7' = [4]7, [3]7' We did not need to list [4],' and [5]7' since, in general, if [a],' = [b]n, then [b],' = [a]n.
In Z12, for example, we have [8]12 congruence classes gives the same answer, no matter which representatives we use: [8]12 + [10]12 = [18]12 = [6]12 and also [20]12 + [34]12 = [54]12 = [6]12. [20]12 and [10]12 = [34]12. Adding
Transcribed Image Text:In Z12, for example, we have [8]12 congruence classes gives the same answer, no matter which representatives we use: [8]12 + [10]12 = [18]12 = [6]12 and also [20]12 + [34]12 = [54]12 = [6]12. [20]12 and [10]12 = [34]12. Adding
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