Question

Cho M 2+ 2 mũ 2+ 2 mũ 3 ........ 2 20. Chứng tỏ rằng M chia hết cho 2;3 và 5

Answer 1

\(M=2\left(1+2+2^2+...+2^{19}\right)⋮2\)

\(M=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{19}\left(1+2\right)=\)

\(=3\left(2+2^3+2^5+...2^{19}\right)⋮3\)

\(M=\left(2+2^3\right)+\left(2^5+2^7\right)+...+\left(2^{17}+2^{19}\right)+\left(2^2+2^4\right)+...+\left(2^{18}+2^{20}\right)\)

\(M=2\left(1+2^2\right)+2^5\left(1+2^2\right)+...+2^{17}\left(1+2^2\right)+...+2^{18}\left(1+2^2\right)\)

\(M=2.5+2^5.5+...+2^{17}.5+...+2^{18}.5⋮5\)