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can you get shingles if you never have had chickenpox - Imagemakers

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 So \( k = 5m + 2 \), \( n = 10(5m + 2) + 6 = 50m + 26 \). Smallest \( n = 26 \). Check: \( 26^2 = 676 \), ends in 76 — valid.

📌 Between 24 and 26, the smaller is 24. Therefore, the smallest positive integer whose square ends in 76 is \( \boxed{24} \).

📌 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}

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