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An epidemiologist tracking a disease outbreak observes that the total number of cases over two consecutive weeks is 144, and the greatest common divisor of the weekly case counts is as large as possible. What is the largest possible value of \( \gcd(a, b) \) if \( a + b = 144 \) and \( a, b \) are positive integers?
Let \( a + b = 144 \), and let \( d = \gcd(a, b) \). Then we can write \( a = d \cdot m \), \( b = d \cdot n \), where \( m \) and \( n \) are coprime positive integers. Then:
a + b = d(m + n) = 144.

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