Problem:
Find the number of sets of three distinct positive integers with the property that the product of , and is equal to the product of and .
Solution:
It is easier to count the ordered triples of positive integers with the product equal to the product . Exactly one of , and is divisible by each of , and . Either one of , and is divisible by or exactly two of , and are divisible by . Hence there are such ordered triples . In three of these ordered triples, two of , and equal . In three of these ordered triples, two of , and equal . In these six cases , and are not distinct. In the remaining ordered triples, , and are distinct. Each of the required unordered triples is represented by of the ordered triples . Therefore the requested number of sets is .
The problems on this page are the property of the MAA's American Mathematics Competitions