Number theory problem

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M F Ruhan
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Number theory problem

Unread post by M F Ruhan » Fri Apr 09, 2021 5:50 pm

x is a random factor of the number 10^99. What is the probability that x is divisible by 10^88?

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Mehrab4226
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Re: Number theory problem

Unread post by Mehrab4226 » Fri Apr 09, 2021 7:04 pm

Total Cases:
Number of factors of $10^{99}=2^{99}\times 5^{99}$ is $100\times 100$
Favorable cases:
All factors which are divisible by $10^{88}$ are in the form, $2^x \times 5^y$ where $x,y$ are integers between $88-99$. So $x$ can have $12$ values and $y$ can have $12$ values, thus the number of factors of that type is $12\times 12$
$\therefore probability=\frac{12\times 12}{100\times 100}=\frac{9}{625}$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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