2020 AIME I Problem 9

Let S be the set of positive integer divisors of 20^{9}. Three numbers are chosen independently and at random with replacement from the set S and labeled a_{1}, a_{2}, and a_{3} in the order they are chosen. The probability that both a_{1} divides a_{2} and a_{2} divides a_{3} is \frac{m}{n}, where m and n are relatively prime positive integers. Find m.