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Home » Dado que ambos son iguales, es un triángulo rectángulo. - Carbonext

Dado que ambos son iguales, es un triángulo rectángulo. - Carbonext

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 $18 = 2 imes 3^2$,

📌 $24 = 2^3 imes 3$,

📌 $36 = 2^2 imes 3^2$.

📌 LCM takes the highest powers: $2^3 imes 3^2 = 8 imes 9 = 72$. Thus, the answer is $oxed{72}$.

📌 Question: What is the largest integer that must divide the product of any three consecutive integers?

📌 Solution: Any three consecutive integers include at least one multiple of 2 and one multiple of 3. Thus, the product is divisible by $2 imes 3 = 6$. Additionally, among three consecutive numbers, there is a multiple of 2 and another even number, ensuring divisibility by $2^2 = 4$ if the sequence includes two even numbers. However, the guaranteed minimum is $3! = 6$. But more precisely, the product of three consecutive integers is divisible by $6$, but not necessarily higher (e.g., $1 imes 2 imes 3 = 6$). Thus, the answer is $oxed{6}$.Question: An ichthyologist models the seasonal variation in coral reef fish populations using the function $ f(x) = \sin^2 x + \cos^2(2x) $. Find the minimum value of $ f(x) $ over all real numbers $ x $.

📌 f(x) = \sin^2 x + \cos^2(2x)

📌 We use trigonometric identities to simplify and analyze the expression. Recall:

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