A=5\(^n\).(5\(^n\)+1)−6\(^n\)(3\(^n\)+2\(^n\))⋮91
A=25\(^n\)+5\(^n\)−18\(^n\)−12\(^n\)\(\left\{{}\begin{matrix}=\left(25^n-18^n\right)-\left(12^n-5^n\right)⋮7\\=\left(25^n-12^n\right)-\left(18^n-5^n\right)⋮13\end{matrix}\right.\Rightarrow A⋮91\)
Đặt \(A=5^n.\left(5^n+1\right)-6^n.\left(3^{n+2}\right)\)
\(\Rightarrow A=\left(25^n-18^n\right)-\left(12^n-5^n\right)\)
Ta có:
\(\left\{{}\begin{matrix}25^n-18^n⋮25-18=7\\12^n-5^n⋮12-5=7\end{matrix}\right.\Leftrightarrow A⋮7\)
Ta lại có:
\(A=\left(25^n-12^n\right)-\left(18^n-5^n\right)\)
Lại có:\(\left\{{}\begin{matrix}25^n-12^n⋮25-12=13\\18^5-5^5⋮18-5=13\end{matrix}\right.\Leftrightarrow A⋮13\)
Mà (7, 13) = 1 và 7 . 13 = 91
\(\Rightarrow A⋮91\)
Vậy \(5^n.\left(5^n+1\right)-6^n\left(3^n+2\right)⋮91\left(đpcm\right)\)