Problem:
When each of and is divided by the positive integer , the remainder is always the positive integer . When each of and is divided by the positive integer , the remainder is always the positive integer . Find .
Solution:
If the remainder is the same when each of and is divided by , then must be a factor of and of . The only common factor of and greater than is , so must be , and the common remainder is . Similarly, must be a factor of and of . The only common factor of and greater than is , so must be , and the common remainder is . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions