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Home » 6 and 9 common multiples - Malaeb

6 and 9 common multiples - Malaeb

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 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, $\phi(48) = \boxed{16}$.

📌 Question: A cartographer detects a triangular layout of ancient ruins where the side lengths are consecutive integers $a-1$, $a$, and $a+1$, and the area is an integer. What is the smallest possible value of $a$ such that the area of the triangle formed is an integer?

📌 Solution: Let the sides be $a-1$, $a$, $a+1$. This is a triangle only if triangle inequalities hold:

📌 $(a-1) + a > a+1 \Rightarrow 2a - 1 > a + 1 \Rightarrow a > 2$.

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