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how old are 7th graders - Imagemakers

📅 April 29, 2026 👤 scraface

Apr 29, 2026

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📌 \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.

📌 Alternatively, note that \( 45 \equiv 0 \mod 9 \), so \( 45^2 \equiv 0 \mod 9 \).

📌 Thus, the remainder is:

📌 An epidemiologist analyzing cumulative case data finds a pattern where the number of cases each day follows a cubic growth model. Find the smallest positive integer \( n \) such that \( n^3 \) ends in the digits 001.

📌 We seek the smallest positive integer \( n \) such that:

📌 n^3 \equiv 1 \pmod{1000}.

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