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Home » \[ A \approx 11592.74 \] - Red Crowns

\[ A \approx 11592.74 \] - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 An entomologist is studying the population dynamics of a certain insect species. The population count, \( n \), is observed to be a three-digit number such that the sum of its digits is a perfect square. What is the largest possible population count?

📌 Let \( n = 100a + 10b + c \), where \( a, b, c \) are digits and \( a

📌 eq 0 \). The sum of the digits is \( a + b + c \), and we require this sum to be a perfect square. The largest three-digit number is 999, so \( a + b + c \) ranges from 1 to 27.

📌 The largest perfect square within this range is \( 25 \) (since \( 5^2 = 25 \)). We seek the largest \( n \) such that \( a + b + c = 25 \).

📌 Given \( a + b + c = 25 \), maximize \( n \) by choosing the largest possible \( a \). Start with \( a = 9 \):

📌 b + c = 25 - 9 = 16

📌 Maximize \( 10b + c \) by setting \( b = 9 \), \( c = 7 \), yielding \( n = 100 imes 9 + 10 imes 9 + 7 = 997 \).

📌 Check: \( 9 + 9 + 7 = 25 \), which is a perfect square. Thus, the largest possible population count is \(oxed{997}\).

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