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a, Ta có:
\(81^7-27^9-9^{13}=3^{28}-3^{27}-3^{26}\)
\(=3^{26}\left(3^2-3-1\right)=3^{25}.3.5=3^{25}.15\)
Vì 15 chia hết cho 15 nên \(3^{25}.15\) chia hết cho 15.
Vậy................(đpcm)
b,Ta có:
\(24^{54}.54^{24}.2^{10}=\left(2^3.3\right)^{54}.\left(2.3^3\right)^{24}.2^{10}\)
\(=2^{162}.3^{54}.2^{24}.3^{72}.2^{10}=2^{196}.3^{126}\)
\(=2^{108}.3^{72}.2^{88}.3^{54}\)
\(72^{36}=\left(2^3.3^2\right)^{36}=2^{108}.3^{72}\)
Vì \(2^{108}.3^{72}\) chia hết cho \(2^{108}.3^{72}\) nên \(2^{108}.3^{72}.2^{88}.3^{54}\) chia hết cho \(2^{108}.3^{72}\)
Vậy............(đpcm)
Chúc bạn học tốt!!!
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. 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)
\(7^6+7^5-7^4\)
\(=7^4\cdot7^2+7^5\cdot7-7^4\)
\(=7^4\cdot\left(7^2+7-1\right)\)
\(=7^4\cdot55\)
\(=7^4\cdot5\cdot11⋮11\left(đpcm\right)\)
\(7^6+7^5-7^4=7^4.\left(7^2+7-1\right)\)
\(=7^4.55⋮11\)
\(=>7^6+7^5-7^4⋮11\)
Ta có: 76 + 75 - 74
= 74 . (49+7-1)
= 74 . 55 chia hết cho 11 => ĐPCM
Ta có: 2454⋅5424⋅210
= (23 . 3)54 . (33 . 2) . 210
= 2162 . 354 . 372. 224 . 210
= 2196 . 3126
= (2189 . 3126). 27
=7263 . 27 chia hết cho 63 => ĐPCM
\(24^{54}.54^{24}.2^{10}\)
\(=\left(2^3.3\right)^{54}.\left(3^3.2\right)^{24}.2^{10}\)
\(=\left(2^3\right)^{54}.3^{54}.\left(3^3\right)^{24}.2^{24}.2^{10}\)
\(=2^{162}.3^{54}.3^{72}.2^{24}.2^{10}\)
\(=2^{196}.3^{126}\)
Lại có :
\(72^{63}=\left(2^3.3^2\right)^{63}\)
\(=\left(2^3\right)^{63}.\left(3^2\right)^{63}\)
\(=2^{189}.3^{126}\)
Vì \(2^{196}.3^{126}⋮2^{189}.3^{126}\Leftrightarrowđpcm\)