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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 A zoologist is studying the migratory patterns of a rare bird species. She notes that the birds return to their nesting grounds every \( n \) years, where \( n \) is the smallest positive integer such that \( n^2 \equiv 1 \pmod{15} \). What is \( n \)?

📌 To solve for \( n \) such that \( n^2 \equiv 1 \pmod{15} \), we start by rewriting the condition as:

📌 n^2 - 1 \equiv 0 \pmod{15} \implies (n-1)(n+1) \equiv 0 \pmod{15}

📌 This means that \( (n-1)(n+1) \) is divisible by 15. Since 15 factors into prime components as \( 3 \times 5 \), both 3 and 5 must divide \( (n-1)(n+1) \).

📌 Step 1: Check modulo 3

📌 The possible residues modulo 3 are 0, 1, and 2:

📌 If \( n \equiv 0 \pmod{3} \), then \( n^2 \equiv 0 \pmod{3} \),

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