SOLUTION: True or False: (Show your solution) a. In modulo 11 is -7^(-1) - 9 = - 8^(-1)? b. In modulo 13 is - 1/7 - 1/3 = - 1/5 ? c. In modulo 11, is 3^(-1) - 1/9 = - 1/8? d.

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Question 1205505: True or False: (Show your solution)
a. In modulo 11 is -7^(-1) - 9 = - 8^(-1)?
b. In modulo 13 is - 1/7 - 1/3 = - 1/5 ?
c. In modulo 11, is 3^(-1) - 1/9 = - 1/8?
d. In modulo 17 is - 8 - 1/7 = - 5^(-1)?
e. In modulo 13 is - 5 - 1/6 = - 6^(-1) ?
Answer by ikleyn(52851)   (Show Source): You can put this solution on YOUR website!
True or False: (Show your solution)
a. In modulo 11 is -7^(-1) - 9 = - 8^(-1)?
b. In modulo 13 is - 1/7 - 1/3 = - 1/5 ?
c. In modulo 11, is 3^(-1) - 1/9 = - 1/8?
d. In modulo 17 is - 8 - 1/7 = - 5^(-1)?
e. In modulo 13 is - 5 - 1/6 = - 6^(-1) ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

(a)  In modulo 11,   is the inverse element to -7 in the multiplicative group Z(11)*.

     It is the same as the inverse element to 4 in the multiplicative group Z(11)*.

     The inverse element to 4 in the multiplicative group Z(11)* is 3  (since 4*3 = 12 = 1 mod 11).

     Therefore,  = 3 in the multiplicative group Z(11)*.



     In modulo 11,   is the inverse element to -8 in the multiplicative group Z(11)*.

     It is the same as the inverse element to 3 in the multiplicative group Z(11)*.

     The inverse element to 3 in the multiplicative group Z(11)* is 4  (since 4*3 = 12 = 1 mod 11).

     Therefore,  = 4 in the multiplicative group Z(11)*.



     Now question (a) is " is it true that  3 - 9 = -4 mod 11 ? "

     Left side is  -6 mod 11.   Right side is -4 mod 11.

     So, equality  " 3 - 9 = -4 mod 11 "  is not true.

     From it, we conclude that the ANSWER to question (a) is  "FALSE".

Do the other questions in the same way, using the same logic.



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