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Problem 807

AMC 12 late, AIME early

Number theory Difficulty 4.5 Find the answer Philippine Mathematical Olympiad · Philippines

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.

Official 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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