Number length?

Number Theory Level 4

How many decimal digits does $2781!$ have?

For example, $5! = 120$ has $3$ decimal digits.

The answer is 8373.

1 solution

Chew-Seong Cheong
Feb 17, 2020

The number of decimal digits of $2781!$ is given by $\left \lfloor \log_{10} 2781! \right \rfloor + 1$ . To estimate $\log_{10} 2781!$ , we can use Stirling's formula as follows:

$\begin{aligned} 2781! & \sim \sqrt{2\pi \cdot 2781} \left(\frac {2781}e \right)^{2781} \\ \log_{10} 2781! & \approx \frac 12 (\log_{10} \pi + \log_{10} 5562) + 2781 (\log_{10} 2781 - \log_{10} e) \\ & \approx 8372.671186 \end{aligned}$

Therefore $2781!$ has $\left \lfloor \log_{10} 2781! \right \rfloor + 1 = \boxed{8373}$ decimal digits.

@Ishtiaque Ahmed Sayef , it is unnecessary to enter everything is LaTex. It is both difficult and not the standard used in Brilliant.org hence does not look professional. I have amended it for you.

Chew-Seong Cheong - 1 year, 3 months ago

Thank you very much. I will try to follow your advice.

Ishtiaque Ahmed Sayef - 1 year, 3 months ago

Master Yoda, "Do. Or do not. There is no try." Please just do it, no try.

Chew-Seong Cheong - 1 year, 3 months ago

Do we even need Stirling's formula? The final formula follows from the scientific notation of numbers.

Atomsky Jahid - 1 year, 3 months ago

Of course we don't need Stirling's formula if we evaluate $\log_{10} 2781!$ by software. We need to evaluate $\displaystyle \sum_{n=1}^{2781} \log_{10} n$ . But this is not a Computer Science problem so I would not use numerical method and Stirling's formula provides a convenient way to solve such problem.

Chew-Seong Cheong - 1 year, 3 months ago

Number length?

The answer is 8373.

1 solution

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