\(S=\left(5+5^4\right)+\left(5^2+5^5\right)+........+\left(5^{2003}+5^{2006}\right)\)
\(S=5\left(1+125\right)+5^2\left(1+125\right)+.........+5^{2003}\left(1+125\right)\)
\(S=126\left(5+5^2+5^3+.........+5^{2003}\right)⋮126\)
Vậy \(S=5+5^2+.........+5^{2006}⋮126\)
\(=\left(5+5^4\right)+\left(5^2+5^5\right)+...+\left(5^{2003}+5^{2006}\right)\)
\(=5\left(1+5^3\right)+5^2\left(1+5^3\right)+...+5^{2003}\left(1+5^3\right)\)
\(=5.126+5^2.126+...+5^{2003}.126\)
\(=126\left(5+5^2+...+5^{2003}\right)\)\(⋮126\)vì\(126⋮126\)
\(\Rightarrow S⋮126\)