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Home » The simplified expression is \( x + 2 \). - Parker Core Knowledge

The simplified expression is \( x + 2 \). - Parker Core Knowledge

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Question: A materials scientist is modeling the crystal lattice of a new self-healing polymer and finds that the number of atoms in a unit cell satisfies a number-theoretic condition: it is a three-digit number that is one less than a multiple of 11 and one more than a multiple of 13. What is the smallest such number?

📌 Solution: We seek the smallest three-digit number $n$ such that:

📌 Use the Chinese Remainder Theorem. Let $n = 13k + 1$. Substitute into the first congruence:

📌 Multiply both sides by the inverse of 2 modulo 11, which is 6:

📌 The smallest three-digit value occurs when $m = 0$: $n = 131$.

📌 Solution: To convert the base-eight number $ 321_8 $ to base-ten, we use the place value expansion:

📌 Solution: The formula for the sum of the first $ n $ squares is:

📌 Solution: A lattice point is a point $ (x, y) $ where both $ x $ and $ y $ are integers. The given equation is:

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