Question

[Tử Đằng]

Cho A= 2²⁰¹¹ +2²⁰¹² +2²⁰¹³ +2²⁰¹⁴ +2²⁰¹⁵ +2²⁰¹⁶

^{Chứng minh A cia hết cho 21}

Answer 1

[Phạm Nguyễn Tất Đạt]

A=\(2^{2011}+2^{2012}+2^{2013}+2^{2014}+2^{2015}+2^{2016}\)

A=\(\left(2^{2011}+2^{2012}\right)+\left(2^{2013}+2^{2014}\right)+\left(2^{2015}+2^{2016}\right)\)

A=\(2^{2011}\left(1+2\right)+2^{2013}\left(1+2\right)+2^{2015}\left(1+2\right)\)

A=\(2^{2011}\cdot3+2^{2013}\cdot3+2^{2015}\cdot3\)

A=\(3\left(2^{2011}+2^{2013}+2^{2015}\right)⋮3\)(1)

A=\(2^{2011}+2^{2012}+2^{2013}+2^{2014}+2^{2015}+2^{2016}\)

A=\(\left(2^{2011}+2^{2012}+2^{2013}\right)+\left(2^{2014}+2^{2015}+2^{2016}\right)\)

A=\(2^{2011}\left(1+2+2^2\right)+2^{2014}\left(1+2+2^2\right)\)

A=\(2^{2011}\cdot7+2^{2014}\cdot7\)

A=\(7\cdot\left(2^{2011}+2^{2014}\right)⋮7\)(2)

Từ (1) và (2)\(\Rightarrow A⋮3,7\)

Mà ƯCLN(3,7)=1

\(\Rightarrow A⋮3\cdot7=21\)