Question

Tìm số dư khi ( 2 mũ 1 + 2 mũ 2 + 2 mũ 3 + 2 mũ 4 + ... + 2 mũ 98 + 2 mũ 99 + 2 mũ 100 ) chia cho 7

Answer 1

\(A=2+2^2+2^3+...+2^{100}\)

\(A=2+\left(2^2+2^3+2^4\right)+...+\left(2^{98}+2^{99}+2^{100}\right)\)

\(A=2+2^2\left(1+2+2^2\right)+...+2^{98}\left(1+2+2^2\right)\)

\(A=2+2^2\cdot7+...+2^{98}\cdot7\)

\(A=2+7\cdot\left(2^2+...+2^{98}\right)\)

Dễ thấy \(7\cdot\left(2^2+...+2^{98}\right)⋮7\)

\(\Rightarrow\) A chia 7 dư 2

Answer 2

Ta thấy

A chia 7 dư 2