\(3n+2⋮2n-1\)
\(\Rightarrow2\left(3n+2\right)-3\left(2n-1\right)⋮2n-1\)
\(\Rightarrow6n+4-6n+3⋮2n-1\)
\(\Rightarrow7⋮2n-1\)
\(\Rightarrow2n-1\inƯ\left(7\right)\left\{1;7\right\}\)
\(\Rightarrow2n\in\left\{2;8\right\}\)
\(\Rightarrow n\in\left\{1;4\right\}\)