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z = -\frac75 - Nelissen Grade advocaten

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \gcd(2^{10} - 1, 2^{15} - 1) = 2^5 - 1 = 32 - 1 = 31

📌 The greatest common divisor is \(\boxed{31}\).

📌 Question:** A zoologist tracking nocturnal animal activity cycles notices behavioral patterns recurring every 9 days, prompting interest in numerical periodicity. What is the smallest three-digit number that is divisible by both 9 and 12?

📌 We seek the least three-digit number divisible by \( \text{lcm}(9, 12) \).

📌 First, compute the LCM:

📌 = 3^2, \quad 12 = 2^2 \times 3 \quad \Rightarrow \quad \text{lcm}(9,12) = 2^2 \times 3^2 = 36

📌 Now, find the smallest three-digit multiple of 36:

📌 \left\lceil \frac{100}{36} \right\rceil = \left\lceil 2.777\ldots \right\rceil = 3

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Using De Moivreâs Theorem:
Here, \( heta = rac{\pi}{6}\), \(n = 6\):
\left( \cos rac{\pi}{6} + i \sin rac{\pi}{6}
ight)^6 = \cos\left(6 \cdot rac{\pi}{6}
ight) + i \sin\left(6 \cdot rac{\pi}{6}
ight) = \cos(\pi) + i \sin(\pi)
Evaluate:
\cos(\pi) = -1, \quad \sin(\pi) = 0

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