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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

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