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Home » Thus, one solution is \( \boxed2\sqrt3 \). - Dachbleche24

Thus, one solution is \( \boxed2\sqrt3 \). - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \( 3^3 = 27 \equiv 3 \)

📌 \( 5^3 = 125 \equiv 5 \)

📌 \( 7^3 = 343 \equiv 7 \)

📌 Only \( n \equiv 1 \pmod{8} \) satisfies \( n^3 \equiv 1 \mod 8 \)? Try \( n = 9 \): \( 9^3 = 729 \equiv 1 \mod 8 \)? \( 729 \div 8 = 91 \times 8 = 728 \), so \( 729 \equiv 1 \). Yes.

📌 But 9 ≡ 1 mod 8, so perhaps only solutions ≡ 1 mod 8?

📌 Wait — try \( n = 1, 9, 17, \dots \), but check modulo 125.

📌 Better: solve \( n^3 \equiv 1 \pmod{1000} \), specifically mod 8 and mod 125.

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