2006 AMC 12A Problems/Problem 8

Problem

How many sets of two or more consecutive positive integers have a sum of ?

Solution

Notice that if the consecutive positive integers have a sum of 15, then their average (which could be a fraction) must be a factor of 15. If the number of integers in the list is odd, then the average must be either 1, 3, or 5, and 1 is clearly not possible. The other two possibilities both work:

• 
• 

If the number of integers in the list is even, then the average will have a . The only possibility is , from which we get:

• 

Thus, the correct answer is C (3).

See also

• 2006 AMC 12A Problems

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Preceded by Problem 7	AMC 12A 2006	Followed by Problem 9