Question

Cho A=2+2²+2³+.....+2¹⁰

a) A chia hết cho3 

b) A chia hết cho 31

Answer 1

a. Ta có: 

\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^9+2^{10}\right)\)

\(=2.\left(1+2\right)+2^3.\left(1+2\right)+...+2^9.\left(1+2\right)\)

\(=2.3+2^3.3+...+2^9.3\)

\(=3.\left(2+2^3+...+2^9\right)\)chia hết cho 3

=> A chia hết cho 3 (đpcm).

b. Ta có:

\(A=\left(2+2^2+2^3+2^4+2^5\right)+\left(2^6+2^7+2^8+2^9+2^{10}\right)\)

\(=2.\left(1+2+2^2+2^3+2^4\right)+2^6.\left(1+2+2^2+2^3+2^4\right)\)

\(=2.\left(1+2+4+8+16\right)+2^6.\left(1+2+4+8+16\right)\)

\(=2.31+2^6.31\)

\(=31.\left(2+2^6\right)\)chia hết cho 31

=> A chia hết cho 31 (đpcm).