Hugely Harmonic

Let NN be the smallest positive integer for which 11+12+13++1N>100.\frac{1}{1}+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{N} > 100. How many digits does NN have?


Note: log10e0.434.\log_{10}e \approx 0.434.

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