Question

Cho . Số dư khi chia cho 31 là

Answer 1

\(A=1+2+2^2+2^3+...+2^{100}\)(có 101 số)

\(A=1+\left(2+2^2+2^3+2^4+2^5\right)+\left(2^6+2^7+2^8+2^9+2^{10}\right)+...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(A=1+2\left(1+2+2^2+2^3+2^4\right)+2^6\left(1+2+2^2+2^3+2^4\right)+...+2^{96}\left(1+2+2^2+2^3+2^4\right)\)

\(A=1+2\cdot31+2^6\cdot31+...+2^{96}\cdot31\)

\(A=1+31\left(2+2^6+...+2^{96}\right)\)

\(\Rightarrow A:31\) dư 1