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) ⋮ 72Câu trả lời: a) ta có: 3 + 32 + 33 + 34 + ... + 399