# A number theory problem by Hemanth K

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, $$8! = 1\times2\times3\times\cdots\times8$$.

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