Answer
No solution in $\mathbb{Z}_{5}$.
Work Step by Step
The elements of $\;\mathbb{Z}_{5}\;$ is 0,1,2,3,4
So, if there is a solution for this equation, then x is one of these numbers.
$If \;\;x=0\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(0)=1 \;\;\; {\color{Red}\rightarrow } \;\;\;\;\;\; 0\neq 1\\
so \;\;0 \;is \;not \;solution\\\\
If \;x=1\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(1)=1\;\; {\color{Red}\rightarrow } \;\;\; 2\neq 1\\
so\; 1\; is\; not\; solution.\\\\
If\;x=2\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(2)=1 \;\;\;\;{\color{Red}\rightarrow }\;\;\;\; 4\neq 1\\
so\; 2\; is\; not\; solution\\\\
If \;x=3\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(3)=1 \;\;\;{\color{Red}\rightarrow } \;\;\;\; 6\neq 1\\
so\; 2\; is\; not\; solution\\\\
If\; x=4\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2.(4)=1 \;\;\;{\color{Red}\rightarrow } \;\;\;8\neq 1\\
so\; 4 \;is\; not\; solution\\\\$
No solution in $\mathbb{Z}_{5}$.
