2017
Problem - 3743
An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
Answer
D
When $N\equiv 0\pmod{5}$, then $N^{16} \equiv 0\pmod{5}$.
When $N\equiv \pm 1\pmod{5}$, then $N^{16} \equiv 1\pmod{5}$.
When $N\equiv \pm 2 \pmod{5}$, then $N^{16} \equiv 2^{16} \equiv 4^8 \equiv (-1)^8\equiv 1\pmod{5}$.
We note that $2020$ is a multiple of $5$. Therefore, the answer is $4/5$.