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Home » = 2^2 \cdot 3, \quad 18 = 2 \cdot 3^2 - Get link 4share

= 2^2 \cdot 3, \quad 18 = 2 \cdot 3^2 - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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

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

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

📌 \( n = 7 \): \( 343 \equiv 7 \mod 8 \)

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