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Home » Thus, the sum of all odd divisors of the number is \(\boxed192\). - Databee Business Systems

Thus, the sum of all odd divisors of the number is \(\boxed192\). - Databee Business Systems

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 w^2 \cdot w - 8w^2 + 9w - 18 = 0 \Rightarrow w^3 - 8w^2 + 9w - 18 = 0.

📌 Wir suchen die Summe der Wurzeln der ursprünglichen Gleichung in \( v \), also der Summe der Quadrate der reellen positiven Wurzeln \( w \), da \( v = w^2 \). Zunächst bestimmen wir die Summe und das Produkt der Wurzeln \( w_i \) des kubischen Polynoms \( w^3 - 8w^2 + 9w - 18 = 0 \). Nach Vieta’s Formeln:

📌 w_1 + w_2 + w_3 = 8, \quad w_1w_2 + w_2w_3 + w_3w_1 = 9, \quad w_1w_2w_3 = 18.

📌 Wir wollen \( \sum v_i = \sum w_i^2 \) berechnen. Mit der Identität:

📌 \sum w_i^2 = (w_1 + w_2 + w_3)^2 - 2(w_1w_2 + w_2w_3 + w_3w_1),

📌 setzen wir die Vieta’schen Werte ein:

📌 \sum w_i^2 = 8^2 - 2 \cdot 9 = 64 - 18 = 46.

📌 Wir müssen überprüfen, ob alle Wurzeln \( w_i \) reell und positiv sind, da nur reelle positive \( w_i \) zu reellen \( v = w_i^2 \) führen. Wir analysieren das kubische Polynom \( f(w) = w^3 - 8w^2 + 9w - 18 \). Berechne Extremstellen:

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