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 $.

Mon, 04 May 2026 17:13:50 +0000

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