Câu hỏi của Trần Hoàng Anh

Question

Chứng tỏ rằng:

a) $3+3^2+3^3+3^4+...+3^{99}⋮13$

b) $81^7-27^9-9^{13}⋮45$

c)$24^{54}.54^{24}.2^{10}⋮72^{63}$

👁💧👄💧👁 · Accepted Answer

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

Câu trả lời:

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

kim chi hàn quốc · Answer

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