Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer Philippines

Problem:
What is the remainder when $3^{2020}$ is divided by $73$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
By Fermat's Little Theorem, $3^{2016} = \left(3^{72}\right)^{28} \equiv 1 \pmod{73}$ . Therefore, $3^{2020} \equiv 3^{4} \equiv 8 \pmod{73}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.