GmatKiss wrote:Is the average of n consecutive integers equal to 1 ?
(1) n is even
(2) if S is the sum of the n consecutive integers, then 0 < S < n

Statement 1:
Important fact: if a set consists of equally-spaced numbers, then the mean and median of the set are equal.
Example: since the numbers in the set {1, 4, 7, 10} are equally spaced (each successive number is 3 greater than the number before it), the mean and median must be equal. Here, the mean and median both equal 5.5
In this question, we have consecutive numbers, and consecutive numbers are equally spaced (each successive number is 1 greater than the number before it), so the mean must equal the median.
So, rather than determine whether or not the mean is equal to 1, we can determine whether or not the median is equal to 1.

Statement 1 tells us that there is an even number of numbers in the set, which means the median (which equals the mean) will be the average of the 2 middlemost numbers.

Of the 2 middlemost numbers, one will be even and one will be odd. So, the median = (odd+even)/2
As you can see, this will result in a median that is not an integer (since odd/2 cannot be an integer). As such, it is impossible for the median (and hence the mean) to equal 1.

So, statement 1 is sufficient.

Statement 2:
If S is the sum of the n numbers, the mean will equal S/n
We are told that 0 < S < n, so if we divide all 3 parts by n, we get:
0/n < S/n < n/n
In other words, 0 < mean < 1
Since we can be certain that the mean cannot equal 1, statement 2 is sufficient and the answer is D.

Cheers,
Brent