\(3n+1⋮11-2n\)
\(\Rightarrow2\left(3n+1\right)⋮11-2n\)
\(\Rightarrow6n+2⋮11-2n\)
\(\Rightarrow-3\left(11-2n\right)+35⋮11-2n\)
\(\Rightarrow35⋮11-2n\)
\(\Rightarrow11-2n\inƯ\left(35\right)=\left\{\pm1;\pm5;\pm7;\pm35\right\}\)
\(\Rightarrow n\in\left\{2;3;5;6;8;9;23;-12\right\}\)