Question

1+2+2^2+2^3+2^4+...+2^14 chia hết cho 31

Answer 1

\(1+2+2^2+2^3+2^4+....+2^{14}\)

\(=\left(1+2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8+2^9\right)+\left(2^{10}+2^{11}+2^{12}+2^{13}+2^{14}\right)\)

\(=\left(1+2+2^2+2^3+2^4\right)+2^5\left(1+2+2^2+2^3+2^4\right)+2^{10}\left(1+2+2^2+2^3+2^4\right)\)

\(=\left(1+2+2^3+2^4\right)\left(1+2^5+2^{10}\right)\)

\(=31\left(1+2^5+2^{10}\right)\)\(⋮\)\(31\)

Answer 2

Đặt A = 1 + 2 + 2² + 2³ + ... + 2¹⁴

=> 2A = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁵

=> 2A - A = 2¹⁵ + 2¹⁴ + ... + 2³ + 2² + 2 - 1 - 2 - 2² - ... - 2¹⁴ 

=> A = 2¹⁵ - 1 

=> A = ( 2⁵ )³ - 1

=> A = 32³ - 1³

Áp dụng hằng đẳng thức a³ - b³ = ( a - b ) ( a² + ab + b² )

=> A = ( 32 - 1 ) ( 32² + 32 +  1 )

=> A = 31 . ( 32² + 33 ) chia hết cho 31

Answer 3

\(1+2^2+2^3+2^4+..+2^{14}=\left(1+2+2^2+2^3+2^4\right)\left(1+2^5+2^{10}\right)\)

\(=31\left(1+2^5+2^{10}\right)⋮31\)