OFFICIAL SOLUTIONF FROM MANHATTANThe problem seeks a quantity that cannot be a sum of the type described. Process of elimination, then, will likely be an efficient solution method. Specifically, if an answer choice can be shown to be the sum of two or more consecutive positive integers, then that answer can be eliminated.

One approach: Test Cases

The problem specifically discusses the sum of two or more consecutive positive integers. Start with the simplest possibility: the sum of two consecutive positive integers.

1 + 2 = 3

2 + 3 = 5

3 + 4 = 7

What’s the pattern? First, any two consecutive positive integers will sum to an add number. Second, any odd sum greater than 1 can be created (the sums will continue to increase in this manner – 3, 5, 7, 9, 11, … – forever).

Therefore, any odd number greater than 1 can be created by adding together two consecutive positive integers. Answers A, C, and E all represent odd numbers; eliminate them.

Try the next simplest case: the sum of three consecutive positive integers.

1 + 2 + 3 = 6

2 + 3 + 4 = 9

3 + 4 + 5 = 12

What’s the pattern here? The sums can be odd or even – no apparent pattern there. Hmm. All three are multiples of 3… is that an actual pattern, though?

Yes, it is! The sum of any set of three consecutive integers can also be calculated by taking the average and multiplying by the number of terms – that is, multiplying by 3. So any sum of three consecutive integers will be a multiple of 3.

As a result, any multiple of 3 (starting with 6) can be created by finding the sum of the appropriate set of three consecutive positive integers. Answer D represents a multiple of 3; eliminate it.

By process of elimination, the only remaining answer is B. The correct answer is B.

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