a. Ta có: \(81^7-27^9-9^{13}\)

\(=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}=3^{28}-3^{27}-3^{26}\)

\(=3^{25}\left(3^3-3^2-3\right)=3^{25}\left(27-9-3\right)=3^{25}\cdot15\)

Vì \(15⋮15\) nên \(3^{25}\cdot15⋮15\)

\(\Rightarrow81^7-27^9-9^{13}⋮15\) (đpcm)

b. Ta có: \(24^{54}\cdot54^{24}\cdot2^{10}\)

\(=\left(2^3\cdot3\right)^{54}\cdot\left(3^3\cdot2\right)^{24}\cdot2^{10}\)

\(=\left(2^3\right)^{54}\cdot3^{54}\cdot\left(3^3\right)^{54}\cdot2^{54}\cdot2^{10}\)

\(=2^{162}\cdot2^{24}\cdot2^{10}\cdot3^{54}\cdot3^{72}=2^{196}\cdot3^{126}\)

Mà \(72^{63}=\left(2^3\cdot3^2\right)^{63}\)

\(=\left(2^3\right)^{63}\cdot\left(3^2\right)^{63}=2^{189}\cdot3^{126}\)

Vì \((2^{196}\cdot3^{126})⋮\left(2^{189}\cdot3^{126}\right)\)

\(\Rightarrow24^{54}\cdot54^{24}\cdot2^{10}⋮72^{63}\) (đpcm)

a, \(81^7-27^9-9^{13}\)

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

\(=3^{22}\left(3^6-3^5-3^4\right)\)

\(=3^{22}\times405⋮405\)

a, \(81^7-27^9-9^{13}\)

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

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

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

\(=3^{25}.3.5\)

\(=3^{25}.15⋮15\)

\(\Leftrightarrow81^7-27^9-9^{13}⋮15\Leftrightarrowđpcm\)

a) \(7^6+7^5-7^4=\left(7^4.7^2\right)+\left(7^3.7^2\right)-\left(7^2.7^2\right)=7^2\left(7^4+7^3+7^2\right)=7^2.1793\)

Mà 1793 chia hết cho 11 => 7².1793 chia hết cho 11

b) \(10^9+10^8+10^7=\left(10^3.10^6\right)+\left(10^2.10^6\right)+\left(10.10^6\right)=10^6.\left(10^3+10^2+10\right)=10^6.1110\)

Mà 1110 chia hết cho 222 => 10⁶.1110 chia hết cho 222

a)

\(7^6+7^5-7^4\)

\(=7^4\cdot\left(7^2+7-1\right)\)

\(=7^4\cdot55⋮55\left(đpcm\right)\)

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c: \(81^7-27^9-9^{13}\)

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

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

\(=3^{24}\cdot45⋮45\)