# A number theory problem by Hemanth K

**Number Theory**Level 3

How many digits are there in the number \( \dfrac{100!}{50!} \) when written in decimal representation?

You may find the following approximations useful.

- \(5! = 120 \) has 3 digits.
- \(\log_{10} 2 \approx 0.3010\).
- \(\log_{10} 3 \approx 0.4771 \).
- \( \log_{10} 7 \approx 0.8451 \).

**Notation**: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).