Problem:
What is the largest positive integer for which there is a unique integer such that
Solution:
By first writing the inequalities in the form , we can see that they are equivalent to
Consequently, the problem is to find the longest open interval that contains exactly one integral multiple of .
Since the length of the above interval is , it contains integers. If , the interval will contain at least two multiples of . Hence, is the largest candidate for . Indeed, we find that
also exhibiting that is the unique positive integer corresponding to .
Note: Even when , it is possible for two integral multiples of 56 to fall within the specified range. Therefore, we verified whether satisfies the condition.
For instance, when , the range from to includes two multiples of 56 — namely and .
The problems on this page are the property of the MAA's American Mathematics Competitions