Question

a) CMR: \(7^6+7^5-7^4⋮55\)

b) CMR: \(3^{n+2}-2^{n+2}+3^n-2^n⋮2\) và \(5\)

c) tìm x:\(\left(x-7\right)^{x+1}=\left(x-7\right)^{x+11}\)

Answer 1

a, Ta có : \(7^6+7^5-7^4\)

\(=7^4.7^2+7^4.7+7^4.1=7^4.49+7^4.7+7^4.1\)

\(=7^4.\left(49+7-1\right)\)

\(=7^4.55\) \(⋮\) \(55\) (vì \(55⋮55\))

Vậy \(7^6+7^5-7^4⋮55\)

b, Ta có : \(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=3^n.3^2-2^n.2^2+3^n-2^n\)

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

\(=3^n.\left(3^2+1\right)-2^n.\left(2^2+1\right)\)

\(=3^n.\left(9+1\right)-2^n.\left(4+1\right)\)

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

\(=3^n.2.5-2^{n-1}.2.5\)

\(=2.5.\left(3^n-2^{n-1}\right)\) chia hết cho 2 và 5( vì \(2⋮2\) ; \(5⋮5\) )

Vậy \(3^{n+2}-2^{n+2}+3^n-2^n\) chia hết cho 2 và 5