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Home » Final answer: \(\boxedf(x) = -\frac14x^3 + \frac114x^2 + 1\) - Dachbleche24

Final answer: \(\boxedf(x) = -\frac14x^3 + \frac114x^2 + 1\) - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: What is the remainder when the sum $ 1^3 + 2^3 + 3^3 + \dots + 10^3 $ is divided by 7?

📌 Solution: We use the formula for the sum of cubes:

📌 \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2

📌 For $ n = 10 $:

📌 \sum_{k=1}^{10} k^3 = \left( \frac{10 \cdot 11}{2} \right)^2 = (55)^2 = 3025

📌 Now compute $ 3025 \mod 7 $. First, reduce 55 modulo 7:

📌 \div 7 = 7 \text{ remainder } 6 \Rightarrow 55 \equiv 6 \pmod{7}

📌 \Rightarrow 55^2 \equiv 6^2 = 36 \equiv 1 \pmod{7}

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