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Home » Unlock Hidden Secrets & Memphis Soundtracks with These 2014 Vibes Classics! - Groen Casting

Unlock Hidden Secrets & Memphis Soundtracks with These 2014 Vibes Classics! - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \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:

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

📌 Use Heron’s formula. Let $s = \frac{(a-1) + a + (a+1)}{2} = \frac{3a}{2}$.

📌 Then area $A = \sqrt{s(s - a+1)(s - a)(s - a-1)} = \sqrt{ \frac{3a}{2} \cdot \left(\frac{3a}{2} - a + 1\right) \cdot \left(\frac{3a}{2} - a\right) \cdot \left(\frac{3a}{2} - a - 1\right)} $

📌 = \sqrt{ \frac{3a}{2} \cdot \left(\frac{a}{2} + 1\right) \cdot \left(\frac{a}{

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