Gọi tổng là S
\(S=1+\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{2007}+5^{2008}\right)\)
\(S=1+5.6+5^3.6+....+5^{2007}.6\)
\(S=1+6.\left(5+5^3+...+5^{2007}\right)\)
Vậy S chia 6 dư 1
\(S=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+....+\left(5^{2006}+5^{2007}+5^{2008}\right)\)
\(S=31.1+31.5^3+....+31.5^{2007}\)
\(S=31.\left(1+5^3+....+5^{2007}\right)\)
Vậy S chia hết cho 31 hay S chia 31 dư 0