Hugely Harmonic

Number Theory Level 3

Let \(N\) be the smallest positive integer for which \[\frac{1}{1}+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{N} > 100.\] How many digits does \(N\) have?

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Note: \(\log_{10}e \approx 0.434.\)