Large Fibonacci Numbers

Since there is only $5 \cdot 5 = 25$ possible remainders that a pair of consecutive Fibonacci numbers can have upon division by $5$ , the sequence of remainders must become periodic with period $\le 25.$

So we can check the remainders of the Fibonacci numbers upon division by 5, which is as follows;

$1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1,0,1,1,....$

We find that the period is $20$ . Hence

$F_{2016}+F_{2017} \equiv F_{16}+F_{17} \equiv 2 + 2 \equiv 4 \pmod 5.$