Tricky problem

Number Theory Level pending

How many decimal digits has the largest number (in base 10) you can represent with 3 digits without additional symbols?

1453678920 3 369693100 4

2 solutions

Laurent Shorts
Apr 7, 2016

Without calculator, we have:

$\log_{10}\left(9^{9^9}\right) = 9^9\log_{10}(9) < 9^9 < 10^9 < 1'453'678'920$ ,

and $\displaystyle 9^9\log_{10}(9) > \sqrt{10}^8\log_{10}(\sqrt{10}) = \frac{10^4}{2} > 4$ .

There's only one possible answer left: 369'693'100.

Thanks for your solution (+1), I use this method many times for solving problems with multiple choices, but we have to be careful with this method...because, has you thought I could be wrong?What would happen then?

Guillermo Templado - 5 years, 2 months ago

I checked afterwards, of course! I was just too lazy at the moment to find or launch a calculator :)

Laurent Shorts - 5 years, 2 months ago

Good,... :)

Guillermo Templado - 5 years, 2 months ago

Guillermo Templado
Apr 7, 2016

The largest number (in base 10) you can represent with 3 digits without additional symbols is $9^{9^9}$ which has $369693100 = \lfloor 9^9 \cdot \log_{10} 9\rfloor + 1$ decimal digits

Tricky problem

2 solutions

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