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a) \(7^6+7^5-7^4=7^4.\left(7^2+7-1\right)=7^4.\left(49+7-1\right)=7^4.55\)
Ta có: 55 chia hết cho 11
Nên \(7^4.55\)chia hết cho 11
Hay \(7^6+7^5-7^4\)chia hết cho 11
Câu b,c làm tương tự
\(24^{54}\cdot54^{24}\cdot2^{10}\)
\(=2^{162}\cdot3^{54}\cdot3^{72}\cdot2^{24}\cdot2^{10}\)
\(=2^{196}\cdot3^{126}\)
a) \(7^6+7^5-7^4=7^4.7^2+7^4.7+7^4.1\)
\(=7^4.\left(7^2+7-1\right)\)
\(=7^4.55\)
Mà \(55⋮11\Rightarrow7^4.55⋮11\Leftrightarrow7^6+7^5-7^4⋮11\left(đpcm\right).\)
b) \(10^9+10^8+10^7=10^6.10^3+10^6.10^2+10^6.10\)
\(=10^6.\left(10^3+10^2+10\right)\)
\(=10^6.1110\)
Mà \(1110⋮222\Rightarrow10^6.110⋮222\Leftrightarrow10^9+10^8+10^7⋮222\left(đpcm\right).\)
c) \(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)
\(=3^{28}-3^{27}-3^{26}\)
\(=3^{26}.3^2+3^{26}.3+3^{26}.1\)
\(=3^{26}.\left(3^2+3+1\right)\)
\(=3^{24}.3^2.5\)
\(=3^{24}.45\)
Mà \(45⋮45\Rightarrow3^{24}.45⋮45\Leftrightarrow81^7-27^9-9^{13}⋮45\left(đpcm\right).\)
d) \(24^{54}.54^{24}.2^{10}=\left(8.3\right)^{54}.\left(27.2\right)^{24}.2^{10}\)
\(=\left(2^3.3\right)^{54}.\left(3^3.2\right)^{24}.2^{10}\)
\(=\left(2^3\right)^{54}.3^{54}.\left(3^3\right)^{24}.2^{24}.2^{10}\)
\(=2^{162}.3^{54}.3^{72}.2^{34}\)
\(=2^{196}.3^{126}\)
\(=2^{189}.2^7.3^{126}\)
\(=\left[\left(2^3\right)^{63}.\left(3^2\right)^{63}\right].2^7\)
\(=\left(8^{63}.9^{63}\right).2^7\)
\(=72^{63}.2^7\)
Mà \(72^{63}⋮72^{63}\Rightarrow72^{63}.2^7⋮72^{63}\Leftrightarrow24^{54}.54^{24}.2^{10}⋮72^{63}\left(đpcm\right).\)
a)
\(7^6+7^5-7^4\)
\(=7^4\cdot\left(7^2+7-1\right)\)
\(=7^4\cdot55⋮55\left(đpcm\right)\)
Mấy câu kia tương tự, dài quá
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
c: \(81^7-27^9-9^{13}\)
\(=3^{28}-3^{27}-3^{26}\)
\(=3^{26}\left(3^2-3-1\right)\)
\(=3^{24}\cdot45⋮45\)