The sum of three positive integers is . Taken in pairs, their differences are , and . Find all possibilities for the three numbers.
Source: NCTM Teacher, December 2005
Suppose the three positive integers are such that . The pairs of difference are and . It is clear that the largest difference belongs to . However, it is not clear how to distribute the two remaining differences and . We have two possible cases to examine.
Case 1: and
not an integer
Case 2: and