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Home » Subtract 2 constant sequences → \(30\) - Londonproperty

Subtract 2 constant sequences → \(30\) - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Frage: Wie viele positive 5-stellige Zahlen sind durch 11 teilbar?

📌 Lösung: Eine Zahl ist durch 11 teilbar, wenn sie die Teilbarkeitsregel für 11 erfüllt, aber statt die einzelnen Zahlen zu prüfen, bestimmen wir die Anzahl der 5-stelligen Zahlen, die durch 11 teilbar sind, mithilfe der Zahlenfolge.

📌 Die kleinste 5-stellige Zahl ist 10000. Wir finden die kleinste 5-stellige Zahl, die durch 11 teilbar ist:

📌 \left\lceil \frac{10000}{11} \right\rceil = \left\lceil 909.\overline{09} \right\rceil = 910, \quad \text{also} \quad 11 \times 910 = 10010.

📌 Die größte 5-stellige Zahl ist 99999. Wir finden die größte durch 11 teilbare Zahl:

📌 \left\lfloor \frac{99999}{11} \right\rfloor = \left\lfloor 9090.8181\ldots \right\rfloor = 9090, \quad \text{also} \quad 11 \times 9090 = 99990.

📌 Die Zahlen, die durch 11 von 10010 bis 99990 teilbar sind, bilden eine arithmetische Folge mit Differenz 11. Die Anzahl der Terme ist:

📌 \frac{99990 - 10010}{11} + 1 = \frac{89980}{11} + 1 = 8180 + 1 = 8181.

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