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Home » Friday Eve Secrets You Won’t Believe Happened at Midnight! - Imagemakers

Friday Eve Secrets You Won’t Believe Happened at Midnight! - Imagemakers

📅 April 29, 2026 👤 scraface

Apr 29, 2026

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📌 Question:** What is the remainder when the sum \( 1^3 + 2^3 + 3^3 + \dots + 12^3 \) is divided by 9?

📌 For \( n = 12 \):

📌 \sum_{k=1}^{12} k^3 = \left( \frac{12 \cdot 13}{2} \right)^2 = (78)^2 = 6084

📌 Now compute \( 6084 \mod 9 \). Use digit sum or direct division:

📌 Sum of digits of 6084: \( 6 + 0 + 8 + 4 = 18 \), and \( 18 \equiv 0 \pmod{9} \), so:

📌 \equiv 0 \pmod{9}

📌 Alternatively, \( 78 \equiv 78 \mod 9 \): \( 7 + 8 = 15 \Rightarrow 1 + 5 = 6 \Rightarrow 78 \equiv 6 \pmod{9} \), so \( 78^2 \equiv 6^2 = 36 \equiv 0 \pmod{9} \).

📌 Thus, the remainder is \( \boxed{0} \).

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