Compute the probability that a randomly chosen positive divisor of is an integer multiple of
Recall previously that the number of divisors is expressed as where each is the exponent of the respective prime in the unique prime factorization of that number. In the previous article, this was established via a simple counting argument on the number of ways of filling out the exponents on the prime factors.
In our problem, it is extremely easy to factor . So . For a divisor of to be a multiple of , it must be the case that its . There are ways of doing this for each exponent, so there are a total of divisors of that are multiples of .
Now, there are a total of divisors of , hence the probability that a randomly chosen divisor of is also a multiple of is .