Question

tìm số dư của A khi chia cho 7 biết

A=1+2²+2³+......... +2²⁰¹¹+2²⁰¹²+2^2013.

Answer 1

Ta có:

\(A=1+2^2+2^3+...+2^{2011}+2^{2012}+2^{2013}\)

\(A=1+\left(2^2+2^3+2^4\right)+\left(2^5+2^6+2^7\right)+...+\left(2^{2011}+2^{2012}+2^{2013}\right)\)

\(A=1+2^2\cdot\left(1+2+2^2\right)+2^5\cdot\left(1+2+2^2\right)+...+2^{2011}\cdot\left(1+2+2^2\right)\)

\(A=1+2^2\cdot7+2^5\cdot7+...+2^{2011}\cdot7\)

\(A=1+7\cdot\left(2^2+2^5+...+2^{2011}\right)\)

Vì \(7⋮7\)

\(\Rightarrow7\cdot\left(2^2+2^5+...+2^{2011}\right)⋮7\)

\(\Rightarrow1+7\cdot\left(2^2+2^5+...+2^{2011}\right)\) chia 7 dư 1

hay \(A\) chia 7 dư 1

Vậy A chia 7 dư 1.