Question

Chứng minh A chia hết cho 14

A = 2+2^2+2^3+.........+2^2019

Answer 1

\(A=2+2^2+2^3+...+2^{2019}\)

   \(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+\left(2^7+2^8+^9\right)+...+\left(2^{2017}+2^{2018}+2^{2019}\right)\) 

   \(=14+2^4\left(2+2^2+2^3\right)+2^7\left(2+2^2+2^3\right)+...+2^{2017}\left(2+2^2+2^3\right)\) 

   \(=14+2^4.14+2^7.14+...+2^{2017}.14\)

   \(=14\left(1+2^4+2^7+...+2^{2017}\right)⋮14\)

    \(\Rightarrow A⋮14\)

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