Question

Cho M = \(2+2^2+2^3+.......+2^{20}.\). Chứng tỏ rằng M \(⋮\)15

Answer 1

Ta có :

\(M=2+2^2+2^3+...+2^{20}\)

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

\(\Rightarrow M=2\left(1+2+2^2+2^3\right)+...+2^{17}\left(1+2+2^2+2^3\right)\)

\(\Rightarrow M=2.15+...+2^{17}.15\)

\(\Rightarrow M=15\left(2+...+2^{17}\right)\)

\(\Rightarrow M⋮15\)

\(\RightarrowĐPCM\)

Answer 2

viết M dưới dạng:

M=\(2.(1+2+2^2+2^3)+2^5.(1+2+2^2+2^3)+...\)

M=\(2.15+2^5.15+...\)

\(=>\) M chia hết cho 15