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Home » V_\textsphere = \frac43 \pi r^3 - Dachbleche24

V_\textsphere = \frac43 \pi r^3 - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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

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

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

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

📌 \( n = 1 \) works.

📌 Actually, solutions are \( n \equiv 1 \pmod{8} \), since the multiplicative group mod 8 has order 4, and \( x^3 \equiv 1 \) has exactly \( \gcd(3, \phi(8)) = \gcd(3,4) = 1 \)? Wait — test more:

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