Question

In $Z_7 - \\{0\\}$ under multiplication mod $7$, if $2^{-1} y \,3^{-1} = 5^{-1}$, then $y =$

• A. 2
• B. 4
• C. 6
• D. none of these

Answer 1

Answer: (B)

4

Answer 2

We know that $Z_7 = \\{1, 2, 3, 4, 5, 6\\}$
Here identity element is 1 and inverse of each element is given by,
$1^{-1} = 1, 2^{-1} = 4, 3^{-1} = 5, 4^{-1} = 2, 5^{-1} = 3, 6^{-1} = 6$
$\therefore \, 2^{-1} y\, 3^{-1} = 5^{-1}$ (given) multiplying by 2 and 3 on left and right respectively
$2\, 2^{-1} \, y\, 3^{-1} 3 = 2\, 5^{-1} 3$
$\Rightarrow \, y=[2?3?3]_{mod \, 7} \Rightarrow \, y=4$