Problem 9 · 2012 Math Kangaroo
Medium
Algebra & Patterns
number-systems
The biggest possible natural number n, for which \(n^{200} < 5^{300}\) holds true, is
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Answer: D — 11
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Hint 1 of 2
Both exponents share a common factor — take a root to simplify the comparison.
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Hint 2 of 2
Reduce to comparing n² with 5³.
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Approach: take the 100th root of both sides
- Take the 100th root of both sides: \(n^{200} < 5^{300}\) becomes \(n^2 < 5^3 = 125\).
- The largest whole \(n\) with \(n^2 < 125\) is 11, since \(11^2 = 121\) but \(12^2 = 144\).
- So the answer is \(n = 11\), choice D.
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