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Home » 2Question: What is the remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12? - Moon Smoking

2Question: What is the remainder when $2025 + 2027 + 2029 + 2031$ is divided by 12? - Moon Smoking

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Solution: This is a stars and bars problem. The formula for distributing $n$ identical items into $k$ distinct boxes is $\binom{n + k - 1}{k - 1}$. Here, $n = 4$, $k = 3$:

📌 \binom{4 + 3 - 1}{3 - 1} = \binom{6}{2} = 15.

📌 The number of ways is $\boxed{15}$.Question: Rationalize the denominator of $\frac{\sqrt{7} - 2}{\sqrt{7} + 3}$ and simplify the result.

📌 Solution: Multiply numerator and denominator by the conjugate of the denominator, $\sqrt{7} - 3$:

📌 \frac{(\sqrt{7} - 2)(\sqrt{7} - 3)}{(\sqrt{7} + 3)(\sqrt{7} - 3)} = \frac{(7 - 3\sqrt{7} - 2\sqrt{7} + 6)}{7 - 9} = \frac{13 - 5\sqrt{7}}{-2} = -\frac{13}{2} + \frac{5\sqrt{7}}{2}

📌 Final answer: $\boxed{-\frac{13}{2} + \frac{5\sqrt{7}}{2}}$

📌 Question: Compute the square of $(2x^2 - 3y)^3$ and simplify the expression.

📌 Solution: First expand $(2x^2 - 3y)^3$ using the binomial theorem:

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Rounded to the nearest cm: <<round(92.5224)=93>>93 cm.

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