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Home » Bubble Letter M Magic: Create Stunning Art You’ll Never Let Go! - Carbonext

Bubble Letter M Magic: Create Stunning Art You’ll Never Let Go! - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \(x = \frac{5\pi}{12} + \pi n\): Solutions are \(\frac{5\pi}{12}, \frac{17\pi}{12}\).

📌 Total of 4 solutions.

📌 Question: If \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, find the maximum value of \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\).

📌 Solution: The dot product \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\). Since \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, each dot product is at most 1. The maximum occurs when \(\mathbf{u}\) aligns with both \(\mathbf{v}\) and \(\mathbf{w}\), i.e., \(\mathbf{v} = \mathbf{w} = \mathbf{u}\). Then \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = 1 + 1 = 2\).

📌 \boxed{2}Question: What is the smallest four-digit number that is a multiple of the least common multiple of the numbers $12$ and $18$, and also divisible by the square of the smallest prime greater than $10$?

📌 First, compute the least common multiple (LCM) of $12$ and $18$.

📌 \text{lcm}(12, 18) = \text{lcm}(2^2 \cdot 3, 2 \cdot 3^2) = 2^2 \cdot 3^2 = 36

📌 The smallest prime greater than $10$ is $11$, so its square is $11^2 = 121$.

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