The sum of consecutive positive integers is . Find all possible values of .

Source: NCTM Mathematics Teacher, December 2005

**SOLUTION**

Let be the sum of consecutive positive integers with the first term equal . If we add the first term to the last term, the second term to the next to the last term, etc.

we notice that they sum to a constant value of . This observation leads us to the formula of the sum

Applying the formula for

Setting equal to one of the divisors of and solving the equation for

The values of for which are positive are , and .

**Answer**: , and .

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