Question

CMR:     a) 81⁷-27⁹-9¹³  chia hết cho 405

              b) 12²ⁿ⁺¹ +11ⁿ⁺²   chia hết cho 133

#ai_pt_giúp_mjk_vs_

Answer 1

       \(81^7-27^9-9^{13}\)         

\(=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)

\(=3^{28}-3^{27}-3^{26}\)

\(=3^{26}\left(3^2-3-1\right)\)

\(=3^{26}.5\)

\(=3^{22}.3^4.5=3^{22}.405⋮405\)

       \(12^{2n+1}+11^{n+2}\)

\(=144^n.12+11^n.121\)

\(=144^n.12-11^n.12+11^n.133\)

\(=\left(144^n-11^n\right).12+11^n.133\)

Ta có: \(a^n-b^n⋮a-b\Rightarrow144^n-11^n⋮133\)

Vậy \(12^{2n+1}+11^{n+2}⋮133\)