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)