1: \(1+5+5^2+\cdots+5^{403}+5^{404}\)
\(=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+\cdots+\left(5^{402}+5^{403}+5^{404}\right)\)
\(=\left(1+5+5^2\right)+5^3\left(1+5+5^2\right)+\cdots+5^{402}\left(1+5+5^2\right)\)
\(=31\left(1+5^3+\cdots+5^{402}\right)\) ⋮31
2: \(1+2+2^2+2^3+\cdots+2^{2001}+2^{2022}\)
\(=\left(1+2\right)+\left(2^2+2^3+2^4\right)+\left(2^5+2^6+2^7\right)+\cdots+\left(2^{2000}+2^{2001}+2^{2002}\right)\)
\(=3+2^2\left(1+2+2^2\right)+2^5\left(1+2+2^2\right)+\cdots+2^{2000}\left(1+2+2^2\right)\)
\(=3+7\cdot\left(2^2+2^5+\cdots+2^{2000}\right)\)
=>\(1+2+2^2+2^3+\cdots+2^{2001}+2^{2022}\) chia 7 dư 3
