Question

In the group $G = \\{1,5,7,11\\}$ under multiplication modulo $12$, the solution of $7^{-1} \; \otimes_{12} (x \otimes_{12} 11) = 5$ is $x$ =

• A. 11
• B. 7
• C. 1
• D. 5

Answer 1

Answer: (C)

1

Answer 2

Given that
$G=\\{1,5,7,11\\}$ is a group under multiplication modulo 12
$\therefore 7^{-1}=7$
$\left(\because 7^{-1} \otimes_{12} 7=1\right)$
Now, $7^{-1} \otimes_{12}\left(x \otimes_{12} 11\right)=5$
$\Rightarrow 7 \otimes_{12}\left(11 \otimes_{12} x\right)=5$
$\Rightarrow \left(7 \otimes_{12} 11\right) \otimes_{12} x=5$
$\Rightarrow 5 \otimes_{12} x=5$
$\Rightarrow x=1$