Discussion

Find the remainder when \(3^{81}\) is divided by 82.

Answer: D

Correct answer: 3

Here \(82 = 2\times 41\) and 3 shares no factor with either, so Euler’s theorem applies.

\[\phi(82) = \phi(2)\,\phi(41) = 1\times 40 = 40 \quad\Longrightarrow\quad 3^{40} \equiv 1 \pmod{82}\]

Since \(81 = 2\times 40 + 1\),

\[3^{81} = \left(3^{40}\right)^{2}\times 3 \equiv 1^{2}\times 3 = 3 \pmod{82}\]

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