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Download Fire Kirin Apk for Android Latest Version - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Thus, the smallest such grid size is $\boxed{1025640}$.

📌 Question: In an online number theory challenge, students are asked: What is the remainder when $3^{2025} + 5^{2025}$ is divided by 8?

📌 Solution: We compute $3^{2025} \mod 8$ and $5^{2025} \mod 8$, then add and reduce mod 8.

📌 First, observe $3^1 = 3$, $3^2 = 9 \equiv 1 \pmod{8}$, so the powers of 3 mod 8 cycle every 2: $3, 1, 3, 1, \dots$

📌 Since $2025$ is odd, $3^{2025} \equiv 3 \pmod{8}$.

📌 Next, $5^1 = 5$, $5^2 = 25 \equiv 1 \pmod{8}$, so powers of 5 mod 8 also cycle every 2: $5, 1, 5, 1, \dots$

📌 $2025$ is odd, so $5^{2025} \equiv 5 \pmod{8}$.

📌 Add: $3 + 5 = 8 \equiv 0 \pmod{8}$.

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