SOLUTION: 2) Determine the order of the following elements. a. 7 in modulo 8 multiplication b. 5 in modulo 11 multiplication

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Question 1205219: 2) Determine the order of the following elements.
a. 7 in modulo 8 multiplication
b. 5 in modulo 11 multiplication
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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Soup of words.


Inedible.


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        A correct mathematical wording should be like  THIS:


(a)   In the multiplicative group   Z/(8Z)^*,  find the order of element   7 (mod 8);

(b)   In the multiplicative group   Z/(11Z)^*,  find the order of element   5 (mod 11).
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                     Solution



(a)  They want you find minimal positive integer n such that 

          7%5En mod 8 = 1.


     Try  n= 1:  7%5E1 mod 8 = 7,  so, n= 1 does not work.

     Try  n= 2:  7%5E2 mod 8 = 49 mod 8 = 48 mod 8 + 1 mod 8 = 0 mod 8 + 1 mod 8 = 1 mod 8.  

     Thus,  n= 2 does work and is the ANSWER.




(b)  They want you find minimal positive integer n such that 

          5%5En mod 11 = 1.


     Try  n= 1:  5%5E1 mod 11 = 5,  so, n= 1 does not work.

     Try  n= 2:  5%5E2 mod 11 = 25 mod 11 = 3 mod 11;   so, n= 2 does not work. 

     Try  n= 3:  5%5E3 mod 11 = (3*5) mod 11 = 15 mod 11 = 4 mod 11;   so, n= 3 does not work. 

     Try  n= 4:  5%5E4 mod 11 = (4*5) mod 11 = 20 mod 11 = 9 mod 11;   so, n= 4 does not work. 

     Try  n= 5:  5%5E5 mod 11 = (9*5) mod 11 = 45 mod 11 = 1 mod 11;   so, n= 5 does work. 

     Thus,  n= 5 does work and is the ANSWER.

Solved.