2017 AMC 10B Problems/Problem 14

Problem

An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probablilty that the remainder when $N^{16}$ is divided by $5$ is $1$ ?

Solution

By Fermat's Little Theorem, $N^{16} = (N^4)^4 \equiv 1 \text{ (mod 5)}$ when N is relatively prime to 5. However, this happens with probability $\boxed{\textbf{(D) } \frac 45}$ .

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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All AMC 10 Problems and Solutions

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2017 AMC 10B Problems/Problem 14

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