S is a set of n consecutive positive integers. Is the mean of the set

Official Magoosh Explanation:

First of all, for a set of consecutive integers, or for any set of evenly spaced numbers, the mean and the median are equal. If there's an odd number of members of the list, then the median is the middle number. If there's an even number of members of the list, then the median is the average of the two middle numbers. For example, the median of {1, 2, 3, 4, 5} is 3, a positive integer and member of the set. For consecutive integers, an even number of members would mean that the mean or median is the average of the two middle integers. For example, the median of {1,2, 3, 4} is the average of 3 and 4, that is, 3.5, not an integer. The only way the mean or median can be an integer is if the set of consecutive integers has an odd number of members.

Statement #1: If there are an even number of consecutive integers, then the evens and odds are balanced in the set, and the first and last number must be opposite: one must be even and the other must be odd. Thus, the range, the difference of (max) – (min) would be either (even) – (odd) or (odd) – (even), in either case, an odd number. If the range is odd, the number of consecutive integers is even.

If there are an odd number of consecutive integers, then the first and last numbers are either both even or both odd. The range would be either (even) – (even) or (odd) – (odd), in either case, an even number. If the range is even, the number of consecutive integers is odd. That must be the case here. As we have seen above, this means the mean or median is a positive integer. This statement, alone and by itself, is sufficient.

Statement #2: As we discussed above, the mean = the median. If the latter is a positive integer, so is the former. This statement, alone and by itself, is sufficient.

Both statement are separately sufficient. Answer = (D)

For a set of positive integers, the mean definitely has to be positive. Is the question right or am I missing something here ?

The mean of a set of n consecutive positive integers will for sure be positive but not necessarily an integer. The mean of a set of n consecutive integers is integer when n is odd, when n is even the mean will be integer/2. For example:

{1, 2, 3} --> mean = 2 = integer;
{1, 2, 3, 4} --> mean = 5/2 = not an integer.

So, the question basically asks whether n is odd.

Bunuel,

Thanks for the response. I certainly didn't pay much attention to the "integer" part which I should have. Now it is clear.

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