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Problem 10 - Entrance Test

How many integers n satisfy 1 ≤ n ≤ 100 and n^3 ≡ 1 (mod 101)?

Correct: A

101 is prime. The equation n³≡1 mod 101 has solutions the cube roots of unity. Since 101≡2 mod 3, the map n↦n³ is bijective, so only n≡1 is a solution.