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Home » We find all \( x \in \2,3,\dots,16\ \) such that \( x^4 \equiv 1 \pmod17 \). - Portal da AcÃºstica

We find all \( x \in \2,3,\dots,16\ \) such that \( x^4 \equiv 1 \pmod17 \). - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: An ornithologist observes a flock of birds migrating, where the number of birds doubles every day starting from 3 birds on day one. On what day does the total count first exceed 500 birds?

📌 Solution: The number of birds follows a geometric sequence: $ 3, 6, 12, 24, \dots $, with first term $ a = 3 $ and common ratio $ r = 2 $.

📌 The number of birds on day $ n $ is $ 3 \cdot 2^{n-1} $.

📌 \cdot 2^{n-1} > 500

📌 2^{n-1} > \frac{500}{3} \approx 166.67

📌 Now compute powers of 2:

📌 $ 2^7 = 128 $, $ 2^8 = 256 $

📌 So $ 2^{n-1} > 166.67 $ implies $ n-1 \geq 8 $, thus $ n \geq 9 $.

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