Ta có:
\(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}\)
\(=3^{n+2}.\left(3^2+1\right)+2^{n+2}.\left(2+1\right)\)
\(=3^{n+1}.10+2^{n+2}.3\)
\(=3^n.3.10+2^{n+1}.2.3\)
\(\Rightarrow3^n.5.6+2^{n+1}.6⋮6\)
\(\Rightarrow3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}⋮6\)
Ta có:
\(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}\)
\(=3^{n+1}.\left(3^2+1\right)+2^{n+2}.\left(2+1\right)=3^{n+1}.10+2^{n+1}.3\)
\(=3^n.10.3+2^n.2.3\)
\(=3^n.30+2^n.6\)
\(=3^n.6.5+2^n.6=6.\left(3^n.5+2^n\right)⋮6\)
Vậy \(3^{n+3}+3^{n+1}+2^{n+3}+2^{n+2}⋮6\)