2017 AMC 10B Problems/Problem 14

Revision as of 13:29, 16 February 2017 by Thedoge (talk | contribs) (Solution)

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 $\boxed{\textbf{(D) } \frac 45}$.

See Also

2017 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All AMC 10 Problems and Solutions

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