A number theory problem by Jun Arro Estrella

Number Theory Level 4

How many digits does $\dfrac{100!}{2^{50}\cdot 50!}$ contain?

The answer is 79.

2 solutions

Christian Daang
Jul 7, 2017

Update:

to find the number of digits, you can just use logarithms.

That is,

$\begin{aligned} \text{number of digits of} \ \dfrac{100!}{2^{50} \cdot 50!} & = \log_{10} \dfrac{100!}{2^{50} \cdot 50!} + 1 \\ & = \log_{10} 100! - \log_{10} 2^{50} - \log_{10} 50! + 1 \\ & = \boxed{79} \end{aligned}$

Jun Arro Estrella
Aug 23, 2015

@Staff: That's 100! Thanks :)

I've updated the problem statement. Can you write up your solution? Thanks!

Brilliant Mathematics Staff - 5 years, 9 months ago

The first thing to do here is to prove that the expression is an integer. Next, Observe that 100!/50!=100P50.. thus, we are left with 100 99 98...51. It can be shown that 2^50 exactly divides the remaining expression, so.. floor(log(ans)+1)=79

Jun Arro Estrella - 5 years, 9 months ago

How 2^50 divides it ??

Chirayu Bhardwaj - 5 years, 6 months ago

@Chirayu Bhardwaj – No of divisors of $2$ in $100!$ is given by $[\frac{100}{2}]+[\frac{100}{4}]+[\frac{100}{8}]+[\frac{100}{16}]+[\frac{100}{32}]+[\frac{100}{64}]$ ,where [.] Is greatest integer function. On calculating it will be $50+25+12+6+3+1$ similarly we can calculate this for $50!$ it will be $25+12+6+3+1$ upon subtracting both we will get $50$ which shows $2^{50}$ is a factor of it...

Atul Shivam - 5 years, 6 months ago

@Atul Shivam – Atul Shivam can u exlpain me again with whole process again pls :)

Chirayu Bhardwaj - 5 years, 6 months ago

Can u please clarify what you have did in last step

Atul Shivam - 5 years, 6 months ago

A number theory problem by Jun Arro Estrella

The answer is 79.

2 solutions

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