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Home » Third term: $ a + d $, first: $ a - 3d $. Ratio: - DevRocket

Third term: $ a + d $, first: $ a - 3d $. Ratio: - DevRocket

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 \(2^3 = 8 \equiv 3\)

📌 \(3^3 = 27 \equiv 2\)

📌 \(4^3 = 64 \equiv 4\)

📌 So only \(n \equiv 2 \pmod{5}\) works.

📌 Now lift to mod 25: Let \(n = 5k + 2\). Plug into \(n^3 \equiv 13 \pmod{25}\):

📌 n^3 = (5k+2)^3 = 125k^3 + 3\cdot25k^2\cdot2 + 3\cdot5k\cdot4 + 8 = 125k^3 + 150k^2 + 60k + 8

📌 60k + 8 \equiv 13 \pmod{25} \Rightarrow 10k + 8 \equiv 13 \pmod{25} \Rightarrow 10k \equiv 5 \pmod{25}

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