Question

B=2^1+2^2+2^3+...+2^30. CMR Bchia hết cho 21

Answer 1

\(B=2+2^2+2^3+...+2^{30}\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+2^5\left(1+2\right)+...+2^{29}\left(1+2\right)\)

\(=2\cdot3+2^3\cdot3+2^5\cdot3+...+2^{29}\cdot3\)

\(=3\left(2+2^3+2^5+...+2^{29}\right)⋮3\)

Mặt khác:\(B=2+2^2+2^3+...+2^{30}\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+....+2^{28}\left(1+2+2^2\right)\)

\(=2\cdot7+2^4\cdot7+....+2^{28}\cdot7\)

\(=7\left(2+2^4+...+2^{28}\right)⋮7\)

Mà (3;7)=1

\(\Rightarrow B⋮3\cdot7=21\)