Bunuel wrote:
The sum of m consecutive integers is 8. If the sum of n consecutive integers is m, what is the value of n?
(A) 15
(B) 16
(C) 31
(D) 32
(E) Cannot be determined by the information provided.
We are given that the sum of m consecutive integers is 8. We see that the m (assuming m > 1) consecutive integers can’t be all positive because we can’t obtain a sum of 8 from m only-positive consecutive integers.. For example, if m = 2, 3 + 4 = 7, and 4 + 5 = 9, and if m = 3, 1 + 2 + 3 = 6, and 2 + 3 + 4 = 9. Thus, some of the integers must be positive and some must be negative, and we see that one of them must, therefore, be 0.
Since the sum is 8, a positive number, there must be more positive integers than negative integers. However, since the negative integers will cancel out their positive counterparts and we have mentioned that there can’t be 2 (or more) consecutive positive integers that add up to 8, there must be exactly 1 more positive integer than the number of negative integers. Furthermore, that positive integer (that won’t cancel with any of its negative counterparts) must be 8. That is, the m consecutive integers must be:
-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8
We see that the sum of all the integers from -7 to 8, inclusive, is 8 since the sum of all the integers from -7 to 7, inclusive, is 0. We also see that there are 16 integers in the above list, so m = 16.
Next we are given that the sum of n consecutive integers is m, which we now know is 16. We can use a similar argument to the one above to conclude that the n (assuming n > 1) consecutive integers can’t be all positive. Furthermore, there must be 1 more positive integer than the number of negative integers and that extra positive integer (the one without a negative counterpart) must be 16. That is, the n consecutive integers must be:
-15, -14, …, -1, 0, 1, …, 14, 15, 16
We see that there are 32 integers in the above list since there are 15 negative integers, 1 zero, and 16 positive integers. Thus, n = 32.
Answer: D