\(S=\left(1+5^2+5^4+5^6\right)+...+\left(5^{2017}+5^{2018}+5^{2019}+5^{2020}\right)\\ S=\left(1+5^2+5^4+5^6\right)+...+5^{2017}\left(1+5^2+5^4+5^6\right)\\ S=\left(1+5^2+5^4+5^6\right)\left(1+...+5^{2017}\right)\\ S=16276\left(1+...+5^{2017}\right)\\ S=52\cdot313\cdot\left(1+...+5^{2017}\right)⋮313\)