Problem:
For how many (not necessarily positive) integer values of is the value of an integer?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because ,
This product will be an integer if and only if both of the factors and are integers, which happens if and only if both exponents are nonnegative. Therefore the given expression is an integer if and only if and . The solutions are exactly the integers satisfying . There are such values.
The problems on this page are the property of the MAA's American Mathematics Competitions