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📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Question: Find the remainder when $ (x^2 + 1)^{2025} + 1 $ is divided by $ x^4 + x^2 + 1 $.

📌 Let $ f(x) = (x^2 + 1)^{2025} + 1 $, and we are to find the remainder when $ f(x) $ is divided by $ x^4 + x^2 + 1 $.

📌 Let $ d(x) = x^4 + x^2 + 1 $. Note that

📌 x^6 - 1 = (x^2 - 1)(x^4 + x^2 + 1),

📌 so $ x^6 \equiv 1 \pmod{d(x)} $. Thus, modulo $ d(x) $, all powers of $ x $ reduce modulo 6.

📌 We reduce $ f(x) \equiv (x^2 + 1)^{2025} + 1 \pmod{d(x)} $.

📌 Letâs analyze $ (x^2 + 1)^{2025} \mod d(x) $.

📌 Let $ y = x^2 $, then $ d(x) = y^2 + y + 1 $. Since $ y^3 \equiv 1 \mod (y^2 + y + 1) $, we have $ y^3 \equiv 1 $, so powers of $ y $ reduce modulo 3.

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