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Time: 2.5 hours - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \div 36 \approx 2.78

📌 So the next integer is 3. Then:

📌 \cdot 3 = 108

📌 is a three-digit number and divisible by both 12 and 18. Thus, $\boxed{108}$ is the smallest such number.

📌 Question: A medical researcher models vaccine efficacy cycles where a booster dose is administered every $\phi(n)$ days for a gene variant with $n = 48$. What is $\phi(48)$, Euler’s totient function value, that determines the periodicity?

📌 Solution: Compute $\phi(48)$. First, factor $48 = 2^4 \cdot 3$. Using the totient formula:

📌 \phi(48) = 48 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 48 \cdot \frac{1}{2} \cdot \frac{2}{3} = 48 \cdot \frac{1}{3} = 16

📌 Thus, $\phi(48) = \boxed{16}$.

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