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Home » between $(0,0,0)$ and $(0,1,1)$: $\sqrt2$, - Crankk.io

between $(0,0,0)$ and $(0,1,1)$: $\sqrt2$, - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 All four satisfy the equation.

📌 Thus, the solutions are $30^\circ, 90^\circ, 150^\circ, 270^\circ$.

📌 Question: Compute $(\cos 15^\circ + i \sin 15^\circ)^6$ and express the result in rectangular form.

📌 Solution: By De Moivre's Theorem:

📌 oxed{i}Question: What is the smallest positive integer whose cube ends in the digits $888$?

📌 Solution: We seek the smallest positive integer $ n $ such that $ n^3 \equiv 888 \pmod{1000} $. This means we want the last three digits of $ n^3 $ to be $888$. We solve this congruence modulo $1000$. Since $1000 = 8 imes 125$, and $8$ and $125$ are coprime, we can use the Chinese Remainder Theorem by solving modulo $8$ and modulo $125$ separately.

📌 First, modulo $8$:

📌 $ n^3 \equiv 888 \pmod{8} $. Since $888 \div 8 = 111$, $888 \equiv 0 \pmod{8}$.

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