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📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 In a study of ancient bird nesting sites, a zoologist models population cycles using a number \( N \) such that \( N \equiv 3 \pmod{4} \), \( N \equiv 2 \pmod{5} \), and \( N \equiv 4 \pmod{7} \). What is the smallest positive \( N \) satisfying all three conditions?

📌 We solve the system of congruences:

📌 N \equiv 3 \pmod{4} \\

📌 N \equiv 2 \pmod{5} \\

📌 N \equiv 4 \pmod{7}

📌 Start with the first two:

📌 Step 1: Solve \( N \equiv 3 \pmod{4} \) and \( N \equiv 2 \pmod{5} \)

📌 Let \( N = 4a + 3 \). Substitute into second:

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Thus, the answer is $ \boxed{576} $.
Question: A geologist analyzing satellite data finds that the elevatio...
Solution: We are given $ a - b = 6 $ and $ a^2 + b^2 = 130 $.
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
We already have $ a - b = 6 $ and $ a^2 + b^2 = 130 $. We need $ ab $.
6^2 = 130 - 2ab \Rightarrow 36 = 130 - 2ab \Rightarrow 2ab = 94 \Right...
Now compute $ a^2 + ab + b^2 = (a^2 + b^2) + ab = 130 + 47 = 177 $.
a^3 - b^3 = 6 \cdot 177 = 1062
Thus, the answer is $ \boxed{1062} $.
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