Question

Cho B = 3+3²+3³+3⁴+3⁵+3⁶+3⁷+3⁸.

Hãy chứng tỏ B chia hết cho 4.

Answer 1

\(B=3+3^2+3^3+3^4+3^5+3^6+3^7+3^8\\=(3+3^2)+(3^3+3^4)+(3^5+3^6)+(3^7+3^8)\\=3\cdot(1+3)+3^3\cdot(1+3)+3^5\cdot(1+3)+3^7\cdot(1+3)\\=3\cdot4+3^3\cdot4+3^5\cdot4+3^7\cdot4\\=4\cdot(3+3^3+3^5+3^7)\)

Vì \(4\cdot(3+3^3+3^5+3^7) \vdots 4\)

nên \(B\vdots4\).

Answer 2

`#3107.101107`

\(B=3+3^2+3^3+3^4+3^5+3^6+3^7+3^8\)

\(=\left(3+3^2\right)+\left(3^3+3^4\right)+\left(3^5+3^6\right)+\left(3^7+3^8\right)\)

\(=3\left(1+3\right)+3^3\left(1+3\right)+3^5\left(1+3\right)+3^7\left(1+3\right)\)

\(=\left(1+3\right)\left(3+3^3+3^5+3^7\right)\)

\(=4\left(3+3^3+3^5+3^7\right)\)

Vì \(4\left(3^3+3^5+3^7\right)\) $\vdots 4$

`\Rightarrow B \vdots 4`

Vậy, `B \vdots 4.`

Answer 3

B=3+32+33+34+35+36+37+38=(3+32)+(33+34)+(35+36)+(37+38)=3⋅(1+3)+33⋅(1+3)+35⋅(1+3)+37⋅(1+3)=3⋅4+33⋅4+35⋅4+37⋅4=4⋅(3+33+35+37)

Vì 4⋅(3+33+35+37)⋮44⋅(3+33+35+37)⋮4

nên $B⋮4$.

Answer 4

B=3+32+33+34+35+36+37+38

=(3+32)+(33+34)+(35+36)+(37+38)=(3+32)+(33+34)+(35+36)+(37+38)

=3(1+3)+33(1+3)+35(1+3)+37(1+3)=3(1+3)+33(1+3)+35(1+3)+37(1+3)

=(1+3)(3+33+35+37)=(1+3)(3+33+35+37)

=4(3+33+35+37)=4(3+33+35+37)

Vì 4(33+35+37)4(33+35+37) $⋮4$

$\Rightarrow B⋮4$

Vậy, $B⋮4.$

Answer 5

Khó