Question

A= 2+2² +2³ +..... +2¹⁰⁰ chứng minh A chia hết cho 15

Answer 1

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

\(\Leftrightarrow A=\left(2+2^2+2^3+2^4\right)+\left(2^4+2^5+2^6+2^7\right)+......+\left(2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(\Leftrightarrow A=2\left(1+2+2^2+2^3\right)+2^4\left(1+2+2^2+2^3\right)+......+2^{97}\left(1+2+2^2+2^3\right)\)

\(\Leftrightarrow A=2.15+2^4.15+.........+2^{97}.15\)

\(\Leftrightarrow A=15\left(2+2^4+.......+2^{97}\right)⋮15\left(đpcm\right)\)