Problem:
The product of three integers is 60 . What is the least possible positive sum of the three integers?
Answer Choices:
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E.
Solution:
Note that , and the sum of these factors is 3. It remains to show that no positive sum can be less than 3 . Such a sum would have to consist of one positive integer and two negative integers with smaller absolute value. If the positive integer is greater than or equal to 10 , then the sum is greater than or equal to 3 . The possible sets of factors in this case are , and . None of these sets of factors has a sum less than .
The only other possible choices for the positive integer are 5 and 6 , and in neither case is a positive sum possible. Indeed, if the positive integer is 5 , then the only possible set of factors is . If the positive integer is 6 , then the only possible set of factors is . In both of these cases, the sum is not positive.
The problems on this page are the property of the MAA's American Mathematics Competitions