2021 Fall AMC 10A Problem 12

Note that $$9 \equiv -1 \pmod{5}.$$ Then if expand $N$ using the definition of bases, we get $$N = 2 \cdot 9^{10} + 7 \cdot 9^9 + 6 \cdot 9^6 +$$$$5 \cdot 9 + 2$$ $$\equiv 2 (-1)^{10} + 7 (-1)^9 + 6 (-1)^6 +$$$$5 (-1) + 2 \pmod{5}$$ $$\equiv 2 - 7 + 6 - 5 + 2 \pmod{5}$$ $$\equiv 3 \pmod{5}.$$ Thus, D is the correct answer.