\(A=2^{100}-2^{99}-...-2^2-2\)

\(2A=2^{101}-2^{100}-...-2^3-2^2\)

\(2A-A=2^{101}-2^{100}-...-2^3-2^2-2^{100}+2^{99}+...2^2+2\)

\(A=2^{101}-\left(2^{100}-2^{100}+2^{99}-2^{99}+...+2^2-2^2+-2\right)\)

\(A=2^{101}+2\)

2¹⁰⁰ - 2⁹⁹ = 2² - 2¹

2¹⁰⁰  -2⁹⁹=2¹

2⁹⁸-2⁹⁷=2¹

=> từ 2¹ -> 2¹⁰⁰ có 50 số 2¹

=>2¹⁰⁰-2⁹⁹-2⁹⁸-...-2¹=2¹.50=2⁵⁰

B= ( 2 + 2^2 + 2^3 + 2^4 + 2^5) + 2^5. ( 2 + 2^2 + 2^3 + 2^4 + 2^5)+....+ 2^95 ( 2 + 2^2 + 2^3 + 2^4 + 2^5)

  = 62.(1 + 2^5 + ... + 2^95 ) chia hết cho 62

Suy ra B chia hết cho 31

 CAM ON NHE DUONG

Ta có; \(A=1+2^2+2^4+...+2^{100}\)

\(\Rightarrow4A=2^2A=2^2+2^4+...+2^{102}\)

\(\Rightarrow3A=4A-A=\left(2^2+...+2^{102}\right)-\left(1+...+2^{100}\right)\)

\(\Rightarrow3A=2^{102}-1\Rightarrow A=\frac{2^{102}-1}{3}\)

tui ko bít =))))