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Home » 1,1,2 → same as 2,1,1 - 500apps

1,1,2 → same as 2,1,1 - 500apps

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 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.

📌 Thus, \( d \) must be a divisor of 144, and \( m + n = \frac{144}{d} \). Since \( m \) and \( n \) are coprime positive integers, \( m + n \geq 2 \), so \( \frac{144}{d} \geq 2 \Rightarrow d \leq 72 \).

📌 To maximize \( d \), we take the largest divisor of 144 such that \( \frac{144}{d} \geq 2 \) and there exist coprime \( m, n \) with \( m + n = \frac{144}{d} \).

📌 The largest divisor of 144 is 144 itself, but then \( m + n = 1 \), impossible. Next, try \( d = 72 \): then \( m + n = 2 \), so \( m = n = 1 \), which are coprime. Then \( a = 72 \), \( b = 72 \), and \( \gcd(72, 72) = 72 \). Valid.

📌 So the maximum possible \( \gcd(a, b) \) is:

📌 \boxed{72}

📌 A hydrologist models contaminant dispersion in groundwater using modular arithmetic, where the signal strength at time step \( t \) modulo 9 determines flow stability. What is the remainder when the sum \( 1^3 + 2^3 + 3^3 + \dots + 9^3 \) is divided by 9?

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