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Home » Can LEGO Wall-E Replace Real Technology? The Cutest Toy Effecting Reality! - Dachbleche24

Can LEGO Wall-E Replace Real Technology? The Cutest Toy Effecting Reality! - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 314_7 = 3 \cdot 7^2 + 1 \cdot 7^1 + 4 \cdot 7^0 = 3 \cdot 49 + 1 \cdot 7 + 4 = 147 + 7 + 4 = 158

📌 \boxed{158}

📌 Question: What is the remainder when $ 2023 + 2025 + 2027 + 2029 $ is divided by 8?

📌 + 2025 + 2027 + 2029 = (2023 + 2029) + (2025 + 2027) = 4052 + 4052 = 8104

📌 Now compute $ 8104 \mod 8 $. Since 8100 is divisible by 8 (as $ 8100 \div 8 = 1012.5 $, but $ 8000 \div 8 = 1000 $, and $ 100 \div 8 = 12.5 $), we check only the last three digits:

📌 \div 8 = 13 \text{ exactly}, so } 8104 \equiv 0 \pmod{8}

📌 $$Question: How many ways are there to distribute 4 distinct chemical samples into 2 identical storage containers such that each container has at least one sample?

📌 Solution: The problem requires counting the number of ways to partition 4 distinct items into 2 non-empty identical subsets. This is given by the Stirling numbers of the second kind, $ S(4, 2) $. The formula for $ S(n, k) $ is $ S(n, k) = S(n-1, k-1) + k \cdot S(n-1, k) $. Using known values, $ S(4, 2) = 7 $. Thus, the number of ways is $ \boxed{7} $.

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