Question

Cho A = \(1+2^1+2^2+2^3+...+2^{2018}\)

Lấy A chia cho 31 thì số dư là?

Answer 1

ta có \(2018:5\) dư \(3\)

\(\Rightarrow A=1+2^1+2^2+2^3+...+2^{2018}\)

\(=1+2^1+2^2+\left(2^3+2^4+2^5+2^6+2^7\right)+...+\left(2^{2014}+2^{2015}+2^{2016}+2^{2017}+2^{2018}\right)\)

\(=7+2^3\left(1+2^1+2^2+2^3+2^4\right)+...+2^{2014}\left(1+2^1+2^2+2^3+2^4\right)\)

\(=7+2^3\left(31\right)+...+2^{2014}\left(31\right)=7+\left(2^3+...+2^{2014}\right).31\)

\(\Rightarrow A\) chia cho \(31\) thì dư \(7\)

vậy \(A\) chia cho \(31\) thì số dư là \(7\)