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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 An epidemiologist analyzing cumulative case data finds a pattern where the number of cases each day follows a cubic growth model. Find the smallest positive integer \( n \) such that \( n^3 \) ends in the digits 001.

📌 We seek the smallest positive integer \( n \) such that:

📌 n^3 \equiv 1 \pmod{1000}.

📌 We solve \( n^3 \equiv 1 \pmod{8} \) and \( n^3 \equiv 1 \pmod{125} \), then use the Chinese Remainder Theorem.

📌 Step 1: Solve \( n^3 \equiv 1 \pmod{8} \)

📌 Try \( n = 1 \): \( 1^3 = 1 \equiv 1 \mod 8 \)

📌 \( n = 3 \): \( 27 \equiv 3 \mod 8 \)

📌 \( n = 5 \): \( 125 \equiv 5 \mod 8 \)

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