2016 AMC 10B 第 8 题

Work modulo $100$ to get the tens digit

Use congruences modulo $4$ and $25$

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 .