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We choose a positive divisor of 20^20 at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of  10^10?

Express your answer as a fraction in simplest form.

 

I know there are 861 divisors of 20^20 and 121 of 10^10 but dont know whats next

 Jul 15, 2021
 #1
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69420 L

 Jul 15, 2021
 #2
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1 - Divisors of 20^20 ==2^40  x  5^20==861 divisors 

 

2 - Divisors of 10^10==2^10  x   5^10==121 divisors

 

3 - Divide (1) by (2) as follows: [2^40 x 5^20] / [2^10 x 5^10] =2^30 x 5^10.

 

4 - The divisors of the answer in (3) are: [30+1] x [10+1] ==31 x 11 ==341 divisors that are multiples of 10^10.

 

5 - Since 20^20 has 861 divisors, the probability that a randomly chosen divisor is a multiple of 10^10 is:

 

341 / 861 ==39.60511034 %

 Jul 16, 2021

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