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Home » 2\sin(z)\cos(z) = \cos(z) - Silent Sales Machine

2\sin(z)\cos(z) = \cos(z) - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: What is the remainder when the sum $2^3 + 4^3 + 6^3 + \dots + 10^3$ is divided by 9?

📌 Solution: Compute the sum of cubes of even numbers from 2 to 10:

📌 $2^3 + 4^3 + 6^3 + 8^3 + 10^3 = 8 + 64 + 216 + 512 + 1000 = 1800$.

📌 Now, divide 1800 by 9: $1800 \div 9 = 200$ with remainder 0. Thus, the remainder is $oxed{0}$.

📌 Question: What is the largest possible value of $\gcd(a, b)$ if $a + b = 100$ and $a, b$ are positive integers?

📌 Solution: The greatest common divisor of $a$ and $b$ must divide their sum $a + b = 100$. The largest divisor of 100 is 50. To achieve $\gcd(a, b) = 50$, set $a = 50$ and $b = 50$. Thus, the maximum value is $oxed{50}$.

📌 Question: What is the least common multiple of 18, 24, and 36?

📌 Solution: Factor each number:

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