Question

\(^{3^{3n+2}-2^{3n+2}+3^{3n}-2^{3n}⋮10}\)

Answer 1

Đặt \(A=3^{3n+2}-2^{3n+2}+3^{3n}-2^{3n}\) ta có : 

\(A=\left(3^{3n+2}+3^{3n}\right)-\left(2^{3n+2}+2^{3n}\right)\)

\(A=\left(3^{3n}.3^2+3^{3n}\right)-\left(2^{3n}.2^2+2^{3n}\right)\)

\(A=3^{3n}\left(3^2+1\right)-2^{3n}\left(2^2+1\right)\)

\(A=3^{3n}.10-2^{3n}.5\)

\(A=5\left(3^{3n}.2-2^{3n}\right)⋮5\) \(\left(1\right)\)

Lại có : 

\(3^{3n}.2⋮2\)

\(2^{3n}⋮2\)

\(\Rightarrow\)\(3^{3n}.2-2^{3n}⋮2\)

\(\Rightarrow\)\(A=5\left(3^{3n}.2+2^{3n}\right)⋮2\) \(\left(2\right)\)

Từ (1) và (2) suy ra \(A\) chia hết cho \(2\) và \(5\)

\(\Rightarrow\)\(A⋮10\) ( đpcm ) 

Vậy \(A⋮10\)

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