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📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 A community nutrition educator is analyzing the distribution of fresh produce in local markets. If two distinct numbers, \( x \) and \( y \), represent the number of units of two types of fresh produce distributed weekly, and it is known that \( x + y = 100 \), what is the largest possible value of \(\gcd(x, y)\)?

📌 Given that \( x + y = 100 \), we aim to find the largest possible \(\gcd(x, y)\). Let \( d = \gcd(x, y) \). Then \( x = dm \) and \( y = dn \) for some integers \( m \) and \( n \) such that \(\gcd(m, n) = 1\). Substituting into the sum, we have:

📌 dm + dn = 100 \implies d(m + n) = 100

📌 To maximize \( d \), \( m + n \) must be minimized. Since \(\gcd(m, n) = 1\), the smallest possible value for \( m + n \) is 2, achieved when \( m = 1 \) and \( n = 1 \). Thus:

📌 d imes 2 = 100 \implies d = rac{100}{2} = 50

📌 Therefore, the largest possible value of \(\gcd(x, y)\) is \(oxed{50}\).

📌 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

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