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new york yankees vs white sox - ClickBalance

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Exponent of $3$: choices $0, 2, 4$ → 3 options

📌 Exponent of $5$: choices $0, 2$ → 2 options

📌 Thus, number of perfect square divisors is $3 \cdot 2 = 6$.

📌 Therefore, there are $ \boxed{6} $ different square grid sizes with integer side lengths that evenly divide the area.

📌 Question: A quantum computing researcher is encrypting data using modular exponentiation in a protocol where the exponent is derived from quadratic residues. What is the remainder when $3^1 + 3^2 + 3^3 + \cdots + 3^{24}$ is divided by $7$?

📌 Solution: We compute the sum $ \sum_{k=1}^{24} 3^k \mod 7 $. This is a geometric series with first term $a = 3$, ratio $r = 3$, and $n = 24$ terms. The sum is:

📌 \sum_{k=1}^{24} 3^k = 3 \cdot \frac{3^{24} - 1}{3 - 1} = \frac{3}{2} (3^{24} - 1)

📌 But working modulo $7$, we can compute the sum directly using cyclicity. First, find the pattern of $3^k \mod 7$:

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