Problem:
How many perfect cubes lie between and , inclusive?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Suppose is any perfect cube in this range. If , then or , which implies .
If , then it follows that , which implies .
This would mean that isn't an integer, leading to a contradiction. Therefore, we know . We also know .
Now suppose . Then we know , which implies , which also means that isn't an integer, leading to another contradiction. Therefore, .
Thus, all which satisfy must also satisfy .
Therefore, the number of possible values for is .
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions