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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) Xét từng vế ta có :
\(24^{54}.54^{24}.2^{10}=\left(2^3.3\right)^{54}.\left(2.3^2\right)^{24}.2^{10}\)
\(=2^{162}.3^{54}.2^{24}.3^{48}.2^{10}\)
\(=2^{172}.3^{102}\)
Xét vế tiếp theo ta có :
\(72^{63}=\left(2^3.3^2\right)^{63}=2^{189}.3^{126}\)
\(\Rightarrow72^{63}⋮24^{54}.2^{10}.54^{24}\)
\(\RightarrowĐPCM\)
a) Ta có: \(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)
\(=3^{28}-3^{27}-3^{26}=3^{22}\left(3^6-3^5-3^4\right)\)
\(=3^{22}\times405\)
\(\Rightarrow81^7-27^9-9^{13}⋮405\)(vì có chứa thừa số 405)
b) Ta có: \(8^7-2^{18}=\left(2^3\right)^7-2^{18}=2^{21}-2^{18}\)
\(=2^{17}\left(2^4-2\right)=2^{17}\times14\)
\(\Rightarrow8^7-2^{18}⋮14\)(vì có chứa thừa số 14)
a, \(10^9+10^8+10^7⋮222\)
Ta có:\(10^9+10^8+10^7=10^7.\left(10^2+10+1\right)\)
\(=10^7.111=5^7.2^7.111=5^7.2^6.2.111=5^7.2^6.222\)
Vì 222\(⋮222\Rightarrow5^7.2^6.222⋮222\)
Vậy \(10^9+10^8+10^7⋮222\)
b) 817 - 279 - 913 ⋮ 45
\(\)Ta có: \(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)
\(=3^{28}-3^{27}-3^{26}=3^{26}.\left(3^2-3-1\right)\)
\(=3^{26}.5=3^{24}.3^2.5=3^{24}.45\)
Vì \(45⋮45\Rightarrow3^{24}.45⋮45\)
Vậy \(81^7-27^9-9^{13}⋮45\)
CHÚC BẠN HỌC TỐT!!
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) 8110 - 2713 - 921 \(⋮\)225
= 8110 - 273 - 921 : 225
= ( 34 )10 - ( 33 )3 - ( 32 )21
= 340 - 39 - 342
= 339 . ( -25 )
= -225 . 337
-225 nhân 1 số tự nhiên thì luôn luôn chia hết cho 225
81^7-27^9-9^13
=(3^4)^7-(3^3)^9-(3^2)^13
=3^28-3^27-3^26
=(3^26.3^2)-(3^26.3^1)-(3^26.1)
=3^26.(9-3-1)
=3^22.(3^4.5)
=3^22.405 chia het cho 405
=> 81^7-27^9-9^13 chia het cho 405
\(81^7-27^9-9^{13}=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)
\(=3^{28}-3^{27}-3^{36}=3^{22}.\left(3^6-3^5-3^4\right)\)
\(=3^{22}.405⋮405\)
a, \(81^7-27^9-9^{13}\)
\(=\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\)
\(=3^{28}-3^{27}-3^{26}\)
\(=3^{26}\left(3^2-3-1\right)\)
\(=3^{25}.3.5\)
\(=3^{25}.15⋮15\)
\(\Leftrightarrow81^7-27^9-9^{13}⋮15\Leftrightarrowđpcm\)