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 =

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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(52848) (Show Source):
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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) ?
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(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.

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.