Problem:
Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let be the probability that exactly one of the selected divisors is a perfect square. The probability can be expressed in the form , where and are relatively prime positive integers. Find .
Solution:
The prime factorization of is . Thus there are 81 positive divisors of . A perfect square divisor of has the form , where each of is either or . Thus there are a total of perfect square divisors of . The requested probability is therefore
and .
The problems on this page are the property of the MAA's American Mathematics Competitions