$%{2017^{2017}} \equiv {4^{2017}} \equiv {2^{4034}} \equiv {2^4} \cdot {\left( {{2^5}} \right)^{806}} \equiv 16 \cdot {\left( {32} \right)^{806}} \equiv 5 \cdot {\left( { - 1} \right)^{806}} \equiv 5$%

5