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chemical equations and balancing - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 \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?

📌 We compute $ S = \sum_{k=1}^{9} k^3 \mod 9 $.

📌 Use the formula for the sum of cubes:

📌 For \( n = 9 \):

📌 \sum_{k=1}^{9} k^3 = \left( \frac{9 \cdot 10}{2} \right)^2 = (45)^2 = 2025.

📌 Now compute \( 2025 \mod 9 \). A quick way: sum the digits of 2025:

📌 + 0 + 2 + 5 = 9 \Rightarrow 2025 \equiv 0 \mod 9.

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