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A=2+2²+2³+...+260A=2+2²+2³+...+260
⇔ A=(2+2²)+...+(259+260)A=(2+2²)+...+(259+260)
⇔ A=2.(1+2)+...+259.(1+2)A=2.(1+2)+...+259.(1+2)
⇔ A=2.3+...+259.3A=2.3+...+259.3
⇔ A=3.(2+..+259)A=3.(2+..+259)
⇒ A⋮ 3
A=2+2²+2³+...+260A=2+2²+2³+...+260
⇔ A=(2+2²+2³)+...+(258+259260)A=(2+2²+2³)+...+(258+259260)
⇔ A=2.(1+2+2²)+...+258.(1+2+2²)A=2.(1+2+2²)+...+258.(1+2+2²)
⇔ A=2.7+...+258.7A=2.7+...+258.7
⇔ A=7.(2+...+258A=7.(2+...+258
⇒ A⋮ 7
Hiện tại mình chưa tìm ra sao chia hết cho 5 nên bạn tự làm nhé cảm ơn bạn
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
Bài 1 :
7^6+7^5-7^4=7^4.49+7^4.7-7^4.1
=7^4.(49+7-1)
=7^4.55
Vì 7^4.55 chia hết 5 Vậy 7^6+7^5-7^4 chia hết 5
a/ \(A=2+2^2+2^3+.....+2^{60}\)
\(=\left(2+2^2\right)+\left(2^3+2^4\right)+.......+\left(2^{59}+2^{60}\right)\)
\(=2\left(1+2\right)+2^3\left(1+2\right)+....+2^{59}\left(1+2\right)\)
\(=2.3+2^3.3+......+2^{59}.3\)
\(=3\left(2+2^3+....+2^{59}\right)⋮3\left(đpcm\right)\)
b/Ta có :
\(A=2+2^2+2^3+.....+2^{60}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+......+2^{58}\left(1+2+2^2\right)\)
\(=2.7+2^3.7+......+2^{58}.7\)
\(=7\left(2+2^3+.....+2^{58}\right)⋮7\left(đpcm\right)\)
c/ \(A=2+2^2+2^3+....+2^{60}\)
\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+....+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+....+2^{57}\left(1+2+2^2+2^3\right)\)
\(=2.15+2^5.15+......+2^{57}.15\)
\(=15\left(2+2^5+......+2^{57}\right)⋮15\left(đpcm\right)\)
A = 2 + 2² + 2³ + ... + 2⁶⁰
= (2 + 2²) + (2³ + 2⁴) + ... + (2⁵⁹ + 2⁶⁰)
= 2.(1 + 2) + 2³.(1 + 2) + ... + 2⁵⁹.(1 + 2)
= 2.3 + 2³.3 + ... + 2⁵⁹.3
= 3.(2 + 2³ + ... + 2⁵⁹) ⋮ 3
Vậy A ⋮ 3
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A = 2 + 2² + 2³ + ... + 2⁶⁰
= (2 + 2² + 2³) + (2⁴ + 2⁵ + 2⁶) + ... + (2⁵⁸ + 2⁵⁹ + 2⁶⁰)
= 2.(1 + 2 + 2²) + 2⁴.(1 + 2 + 2²) + ... + 2⁵⁸.(1 + 2 + 2²)
= 2.7 + 2⁴.7 + ... + 2⁵⁸.7
= 7.(2 + 2⁴ + ... + 2⁵⁸) ⋮ 7
Vậy A ⋮ 7
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A = 2 + 2² + 2³ + ... + 2⁶⁰
= (2 + 2² + 2³ + 2⁴) + (2⁵ + 2⁶ + 2⁷ + 2⁸) + ... + (2⁵⁷ + 2⁵⁸ + 2⁵⁹ + 2⁶⁰)
= 30 + 2⁴.(2 + 2² + 2³ + 2⁴) + ... + 2⁵⁶.(2 + 2² + 2³ + 2⁴)
= 30.(1 + 2⁴ + ... + 2⁵⁶)
= 5.6.(1 + 2⁴ + ... + 2⁵⁶) ⋮ 5
Vậy A ⋮ 5
\(A=2+2^2+2^3+...+2^{60}\)
\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)
\(A=6+2^2.\left(2+2^2\right)+...+2^{58}.\left(2+2^2\right)\)
\(A=6+2^2.6+...+2^{58}.6\)
\(A=6.\left(1+2^2+...+2^{58}\right)\)
Vì \(6⋮3\) nên \(6.\left(1+2^2+...+2^{58}\right)⋮3\)
Vậy \(A⋮3\)
___________
\(A=2+2^2+2^3+...+2^{60}\)
\(A=\left(2+2^2+2^3\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)
\(A=14+...+2^{57}.\left(2+2^2+2^3\right)\)
\(A=14+...+2^{57}.14\)
\(A=14.\left(1+...+2^{57}\right)\)
Vì \(14⋮7\) nên \(14.\left(1+...2^{57}\right)⋮7\)
Vậy \(A⋮7\)
____________
\(A=2+2^2+2^3+...+2^{60}\)
\(A=\left(2+2^2+2^3+2^4\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(A=30+...+2^{56}.\left(2+2^2+2^3+2^4\right)\)
\(A=30+...+2^{56}.30\)
\(A=30.\left(1+...+2^{56}\right)\)
Vì \(30⋮5\) nên \(30.\left(1+...+2^{56}\right)⋮5\)
Vậy \(A⋮7\)
\(#WendyDang\)