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

Guest Jul 15, 2021

#2**+1 **

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 %**

Guest Jul 16, 2021