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Number Theory Level 4

1 0 n < n ! \Large{ 10^n < n! }

Find the smallest natural number n n for which the above inequality satisfies.

You may use a logarithmic table for your calculation.

Inspiration .


The answer is 25.

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Satyajit Mohanty by Satyajit Mohanty , 24, India

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3 solutions

Just try to solve the inequality

n < log ⁡ 10 n ! = ∑ i = 2 n log ⁡ 10 i n < \log_{10} n! = \sum_{i=2}^n \log_{10} i

3 Helpful 0 Interesting 0 Brilliant 0 Confused

24 ! − 1 0 24 = − 3.7.... E 23 < 0 25 ! − 1 0 25 = 5.5..... E 24 > 0 24!-10^{24}=-3.7....E23<0 \\ 25!-10^{25}=5.5.....E24>0

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Harshi Singh
Aug 16, 2015

even i don't know inequality ...i just solve it by assumption can anybody tell me its method....??

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Would you want a C++ solution? For that is how I did it. What about you @Satyajit Mohanty

User 123 - 5 years, 10 months ago

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