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Problem 6 - Entrance Test

Let S be the set of all positive integers n ≤ 1000 such that n^2 ≡ 1 (mod 1000). Find |S|.

Correct: B

1000=8·125. CRT: solve n²≡1 mod 8 and mod 125. Mod 8: n≡±1,±3 (4 solutions). Mod 125: n²≡1 ⇒ n≡±1 (2 solutions). Total solutions=4·2=8.