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a) \(\left(1+2+2^2+...+2^7\right)\)
\(=\left(1+2\right)+\left(2^2+2^3\right)+...+\left(2^6+2^7\right)\)
\(=\left(1+2\right)+2^2.\left(1+2\right)+...+2^6.\left(1+2\right)\)
\(=3+2^2.3+...+2^6.3\)
\(=3.\left(1+2^2+...+2^6\right)⋮3\left(đpcm\right)\)
a) Đặt A = 1 + 2 + 22 + 23 + ... + 27
Ta có:
A = 1 + 2 + 22 + 23 + ... + 27
\(\Rightarrow\)2A = 2 + 22 + 23 + 24 + ... + 28
\(\Rightarrow\)A = 28 - 1 = 255
Vì 255\(⋮\)3\(\Rightarrow\)2 + 22 + 23 + 24 + ... + 28\(⋮\)3
\(\Rightarrow\)ĐPCM
1)
a)\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)
Vì \(3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)chia hết cho 3 nên \(B⋮3\)
\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=\left(3+3^3+3^5+3^7\right)+.....+\left(3^{1988}+3^{1989}+3^{1990}+3^{1991}\right)\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6\right)+.....+3^{1988}\left(1+3^2+3^4+3^6\right)\)
\(\Leftrightarrow B=3.820+.....+3^{1988}.820\)
\(\Leftrightarrow B=3.20.41+.....+3^{1988}.20.41\)
Vì \(3.20.41+.....+3^{1988}.20.41\) chia hết cho 41 nên \(B⋮41\)
A=2+2^2+...........+2^60
c\m c\h cho 3:2+2^2+....+2^60=2.(1+2)+........+2^59(1+2)
=2.3+.........+2^59.3
=(2+...+2^59).3
=>A chia hết cho 3
cau tiếp tuong tu
3
Ta chứng minh A chia hết cho 3:
A=(2+2^2)+(2^3+2^4)+...+(2^59+2^60)
=2.(1+2)+2^3.(1+2)+...+2^59.(1+2)
=2.3+2^3.3+...+2^59.3
=3.(2+2^3+...+2^59) chia hết cho 3
Ta chứng minh A chia hết cho 7
A=(2+2^2+2^3)+(2^4+2^5+2^6)+...+(2^58+2^59+2^60)
=2.(1+2+4)+2^4.(1+2+4)+...+2^58.(1+2+4)
=2.7+2^4.7+...+2^58.7
=7.(2+2^4+...+2^58) chia hết cho 7
Ta chứng minh A chia hết cho 15
A=(2+2^2+2^3+2^4)+(2^5+2^6+2^7+2^8)+...+(2^57+2^58+2^59+2^60)
=2.(1+2+4+8)+2^5.(1+2+4+8)+....+2^57.(1+2+4+8)
=2.15+2^5.15+..+2^57.15
=15.(2+2^5+...+2^57) chia hết cho 15
a) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)
\(=\left(3+3^3+3^5\right)+\left(3^7+3^9+3^{11}\right)+...+\left(3^{1987}+3^{1989}+3^{1991}\right)\)
\(=3\times\left(1+3^2+3^4\right)+3^7\times\left(1+3^2+3^4\right)+...+3^{1987}\times\left(1+3^2+3^4\right)\)
\(=3\times91+3^7\times91+...+3^{1987}\times91\)
\(=3\times7\times13+3^7\times7\times13+...+3^{1987}\times7\times13\)
\(=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)
Vì \(A=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)nên A chia hết cho 13.
b) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)
\(=\left(3+3^3+3^5+3^7\right)+...+\left(3^{1985}+3^{1987}+3^{1989}+3^{1991}\right)\)
\(=3\times\left(1+3^2+3^4+3^6\right)+...+3^{1985}\times\left(1+3^2+3^4+3^6\right)\)
\(=3\times820+...+3^{1985}\times820\)
\(=3\times20\times41+...+3^{1985}\times20\times41\)
\(=41\times\left(3\times20+...+3^{1985}\times20\right)\)
Vì \(A=41\times\left(3\times20+...+3^{1985}\times20\right)\)nên A chia hết cho 41.
a) \(C=5+5^2+5^3+...+5^8\)
\(C=\left(5+5^2\right)+\left(5^3+5^4\right)+\left(5^5+5^6\right)+\left(5^7+5^8\right)\)
\(C=\left(5+25\right)+5^2\cdot\left(5+25\right)+5^4\cdot\left(5+25\right)+5^6\cdot\left(5+25\right)\)
\(C=30+5^2\cdot30+5^4\cdot30+5^6\cdot30\)
\(C=30\cdot\left(1+5^2+5^4+5^6\right)\)
Vậy C chia hết cho 30
b) \(D=2+2^2+2^3+...+2^{60}\)
\(D=2\left(1+2\right)+2^2\left(1+2\right)+...+2^{59}\cdot\left(1+2\right)\)
\(D=2\cdot3+2^2\cdot3+...+2^{59}\cdot3\)
\(D=3\cdot\left(2+2^2+...+2^{59}\right)\)
Vậy D chia hết cho 3
\(D=2+2^2+2^3+...+2^{60}\)
\(D=2\cdot\left(1+2+4\right)+2^4\cdot\left(1+2+4\right)+...+2^{58}\cdot\left(1+2+4\right)\)
\(D=2\cdot7+2^4\cdot7+...+2^{58}\cdot7\)
\(D=7\cdot\left(2+2^4+...+2^{58}\right)\)
Vậy D chia hết cho 7
\(D=2+2^2+2^3+...+2^{60}\)
\(D=\left(2+2^2+2^3+2^4\right)+....+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)
\(D=2\cdot\left(1+2+4+8\right)+...+2^{57}\cdot\left(1+2+4+8\right)\)
\(D=2\cdot15+2^5\cdot15+...+2^{57}\cdot15\)
\(D=15\cdot\left(2+2^5+...+2^{57}\right)\)
Vậy D chia hết cho 15
a) C = 5 + 5² + 5³ + ... + 5⁸
= (5 + 5²) + 5².(5 + 5²) + 5⁴.(5 + 5²) + 5⁶.(5 + 5²)
= 30 + 5².30 + 5⁴.30 + 5⁶.30
= 30.(1 + 5² + 5⁴ + 5⁶) ⋮ 30
Vậy C ⋮ 30
b) *) Chứng minh D ⋮ 3
D = 2 + 2² + 2³ + ... + 2⁶⁰
= 2.(1 + 2) + 2³.(1 + 2) + ... + 2⁵⁹.(1 + 2)
= 2.3 + 2³.3 + ... + 2⁵⁹.3
= 3.(2 + 2³ + ... + 2⁵⁹) ⋮ 3
Vậy D ⋮ 3 (1)
*) Chứng minh D ⋮ 7
D = 2 + 2² + 2³ + ... + 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 D ⋮ 7 (2)
*) Chứng minh D ⋮ 15
D = 2 + 2² + 2³ + ... + 2⁶⁰
= 2.(1 + 2 + 2² + 2³) + 2⁵.(1 + 2 + 2² + 2³) + 2⁵⁷.(1 + 2 + 2² + 2³)
= 2.15 + 2⁵.15 + ... + 2⁵⁷.15
= 15.(2 + 2⁵ + ... + 2⁵⁷) ⋮ 15
Vậy D ⋮ 15 (3)
Từ (1), (2), (3) suy ra D chia hết cho lần lượt 3; 7 và 15
1) \(B=1+5+5^2+5^3+....+5^{101}\)
\(=\left(1+5\right)+\left(5^2+5^3\right)+.....+\left(5^{100}+5^{101}\right)\)
\(=\left(1+5\right)+5^2\left(1+5\right)+....+5^{100}\left(1+5\right)\)
\(=\left(1+5\right)\left(1+5^2+....+5^{100}\right)\)
\(=6\left(1+5^2+...+5^{100}\right)\)\(⋮6\)
2) \(C=81^3+3^{14}+27^5\)
\(=\left(3^4\right)^3+3^{14}+\left(3^3\right)^5\)
\(=3^{12}+3^{14}+3^{15}\)
\(=3^{12}.\left(1+3^2+3^3\right)\)
\(=3^{12}.37\)\(⋮37\)