Problem:
If are consecutive positive integers such that is a perfect square and is a perfect cube, what is the smallest possible value of
Solution:
Since and are consecutive integers, and . Let and be positive integers such that and . Then
and
From we see that and hence . (If and are integers, is read" divides " and means that is a factor of .) Therefore, implies that and hence that . From we also find that , which leads to . Consequently, divides c. It is easy to verify that is the solution we seek.
The problems on this page are the property of the MAA's American Mathematics Competitions