2016 AMC 10B Problem 8

To find the tens digit, we first need to find the number modulo $100.$ First, we can find $2015^{2016} \mod 100.$ We can do this with the Chinese Remainder Theorem by first getting the number $\mod 4$ and then $\mod 25.$ The number $$2015^{2016} \equiv 0 \mod 25$$ since it is a multiple of $25.$ Then, observe that $$2015^{2016} \equiv (-1)^{2016}$$$$\equiv 1 \mod 4 .$$ By the Chinese remainder theorem, we can get that our number is congruent to $25 \mod 100.$ Therefore, $$2015^{2016}-2017$$$$\equiv 25 - 17 \mod 100$$$$\equiv 8 \mod 100 .$$ This means the tens digit is $0.$ Thus, the correct answer is A .