Question

[✎﹏🅷ạ🅽🅷︵❣🅿🅷ú🅲︵❣Đé🅾︵❣🅲ó︵❣Đâ🆄︵❣✔]

cho A = 2+2²+2³+...+2⁶⁰. chứng minh A\(⋮\)3,7,15

Answer 1

[Nhật Hạ]

\(A=2+2^2+2^3+...+2^{60}\)

\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)

\(A=2\cdot\left(1+2\right)+2^3\cdot\left(1+2\right)+...+2^{59}\cdot\left(1+2\right)\)

\(A=2\cdot3+2^3\cdot3+...+2^{59}\cdot3\)

\(A=3\cdot\left(2+2^3+...+2^{59}\right)\)

\(\Rightarrow A⋮3\)

\(A=2+...+2^{60}\)

\(A=\left(2+2^2+2^3\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(A=2.\left(1+2+2^2\right)+...+2^{58}.\left(1+2+2^2\right)\)

\(A=2.7+...+2^{58}.7\)

\(A=7.\left(2+...+2^{58}\right)\)

\(\Rightarrow A⋮7\)

\(A=2+2^2+...+2^{60}\)

\(A=\left(2+2^2+2^3+2^4\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(A=2.\left(1+2+2^2+2^3\right)+...+2^{57}.\left(1+2+2^2+2^3\right)\)

\(A=2.15+...+2^{57}.15\)

\(A=15.\left(2+...+2^{57}\right)\)

\(\Rightarrow A⋮15\)