Problem:
For how many three-element sets of positive integers is it true that
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the three factors, and , must be distinct, we seek the number of positive integer solutions to
The prime factors of and must be disjoint subsets of , no more than one subset can be empty, and the union of the subsets must be . The numbers of elements in the subsets can be: ; or .
In the case, there are ways to choose three subsets with these sizes. In the case, there are ways to choose the three subsets.
In the case, there are ways to choose the three subsets.
In the case, there are ways of choosing the one-element subset and ways of dividing the remaining four elements into two subsets of two elements each, yielding ways of choosing the three subsets in this case.
Thus there are a total of ways of choosing our three subsets and, therefore, ways of expressing in the required manner. Since factorization into primes is unique, these triplets of sets give distinct solutions.
There are ordered triples, , of integers such that , since each of the five prime factors of divides exactly one of or . In three of these ordered triples, two of equal . In the remaining ordered triples, and are distinct, since is square-free. Each unordered triple whose product is is represented by of the ordered triples , so the answer is .