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Home » 2**Question:** Find the smallest positive integer whose square ends in 76. - Crankk.io

2Question: Find the smallest positive integer whose square ends in 76. - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \min |2\sin x + 3\cos x + 4| = \min (2\sin x + 3\cos x + 4) = 4 - \sqrt{13} + 0? \quad \text{Wait — no: the minimum of the inner expression is $ 4 - \sqrt{13} \approx 0.395 $, so the minimum of the absolute value is just its minimum value because it's positive.

📌 \min |2\sin x + 3\cos x + 4| = 4 - \sqrt{13} \quad \text{if } 4 - \sqrt{13} > 0

📌 But $ 4 - \sqrt{13} > 0 $ since $ \sqrt{13} < 4 $, so yes.

📌 But wait: is this the minimum? Since the expression inside is always greater than $ 0 $, the absolute value is minimized exactly when $ 2\sin x + 3\cos x $ is minimized.

📌 \min (2\sin x + 3\cos x + 4) = 4 + \min(2\sin x + 3\cos x) = 4 + (-\sqrt{13}) = 4 - \sqrt{13}

📌 And since this value is positive, the absolute value does not introduce a smaller value.

📌 Thus, the minimum value is $ \boxed{4 - \sqrt{13}} $.

📌 Question: An anthropologist studying ritualistic circular dance formations interprets rhythmic movements as vectors; find the vector $ \mathbf{v} = \langle x, y, z \rangle $ satisfying $ \mathbf{v} \times \langle 1, 2, 3 \rangle = \langle 1, -2, 1 \rangle $ and $ \mathbf{v} \cdot \langle 1, 2, 3 \rangle = 0 $.

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