Question

a)Chứng minh rằng mọi n nguyên dương đều có :5^n(5^n+1)-6^n(3^n+2)chia hết cho 91

Answer 1

\(91=7.13\)

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:

\(\left\{\begin{matrix}25^n-18^n⋮25-18=7\\12^n-5^n⋮12-5=7\end{matrix}\right.\)\(\Leftrightarrow A⋮7\)

Mặt khác:

\(A=\left(25^n-12^n\right)-\left(18^n-5^n\right)\)

Lại có:

\(\left\{\begin{matrix}25^n-12^n⋮25-12=13\\18^n-5^n⋮18-5=13\end{matrix}\right.\)\(\Leftrightarrow A⋮13\)

Mà: \(\left(7;13\right)=1\)

\(\Leftrightarrow A⋮91\)

Vậy \(5^n\left(5^n+1\right)-6^n\left(3^{n+2}\right)⋮91\) (Đpcm)