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) Có: \(3+3^2+3^3+3^4+...+3^{99}\\ =\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{97}+3^{98}+3^{99}\right)\\ =\left(3+3^2+3^3\right)+3^3\left(3+3^2+3^3\right)+...+3^{97}\left(3+3^2+3^3\right)\\ =39+3^3\cdot39+...+3^{97}\cdot39\\ =13\cdot3+3^3\cdot13\cdot3+...+3^{97}\cdot13\cdot3\\ =13\left(3+3^4+...+3^{98}\right)⋮13\left(đpcm\right)\)

b) Có: \(81^7-27^9-9^{13}\\ =\left(3^4\right)^7-\left(3^3\right)^9-\left(3^2\right)^{13}\\ =3^{28}-3^{27}-3^{26}\\ =3^{26}\left(3^2-3-1\right)\\ =3^{24}\cdot\left(3^2\cdot5\right)\\ =3^{24}\cdot45⋮45\left(đpcm\right)\)

c) Có: \(24^{54}\cdot54^{24}\cdot2^{10}\\ =\left(2^3\cdot3\right)^{54}\cdot\left(2\cdot3^3\right)^{24}\cdot2^{10}\\ =2^{162}\cdot3^{54}\cdot2^{24}\cdot3^{72}\cdot2^{10}\\ =2^{196}\cdot3^{126}\\ =2^7\cdot\left(2^{189}\cdot3^{126}\right)\\ =2^7\cdot\left[\left(2^3\right)^{63}\cdot\left(3^2\right)^{63}\right]\\ =2^7\left(8^{63}\cdot9^{63}\right)\\ =2^7\cdot72^{63}⋮72^{63}\left(đpcm\right)\)

a) ta có: 3 + 3² + 3³ + 3⁴ + ... + 3⁹⁹

= (3 + 3² + 3³) + (3⁴ + 3⁵ +36) + ... + (3⁹⁷ + 3⁹⁸ + 3⁹⁹)

= 3(1 + 3 + 3²) + 3⁴(1 + 3 + 3) + ... + 3⁹⁶(1 + 3 + 3)

= 3.13 + 3⁴.13 + ... + 3⁹⁶.13

= 13(3 + 3⁴ + ... + 3⁹⁶) ⋮ 13

vậy (3 + 3² + 3³ + 3⁴ + ... + 3⁹⁹) ⋮ 13

b) ta có: 81⁷ - 27⁹ - 9¹³

= (3⁴)⁷ - (3³)⁹ - (3²)¹³

= 3²⁸ - 3²⁷ - 3²⁶

= 3²⁶(3² - 3 - 1)

= 3²⁶ . 5 = 3²⁴ (9.5) = 3²⁴ . 45 ⋮ 45

Vậy (81⁷ - 27⁹ - 9¹³) ⋮ 45

c) ta có: 24⁵⁴.54²⁴.2¹⁰

= (2³.3)⁵⁴ . (2.3³)²⁴ . 2¹⁰

= 2¹⁶² . 3⁵⁴ . 2²⁴ . 3⁷² . 2¹⁰

= 2¹⁹⁶ . 3¹²⁶

= (2¹⁹³.3¹²⁴).(2³.3²)

= (2¹⁹³.3¹²⁴).72 ⋮ 72

vậy (24⁵⁴.54²⁴.2¹⁰) ⋮ 72

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

chứng minh nó

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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