Question: 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

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

We are asked if the mean of n consecutive is an integer or not.

N consecutive integer
This implies Evenly spaced set =>Mean = Median =Average of the first and the last term

Mean for N consecutive integers can take only 2 forms =>
Let x be N = odd
Let x.5 be for N=even
for some integer x.

Statement 1

N is even
So mean will never be an integer(Because median will never be an integer)
Hence meaning can never be "one".
Hence Sufficient

Statement 2

Let Sum be S
And 0 < S < N

Mean = \frac{Sum}{x}
Mean = S/N
As 0 < S < N

S/N will be a decimal between (0,1)
This mean it will never be one, Sufficient
Hence the correct answer is D.

Approach Solution 2:

To get an average=1; the Sum of elements must be equal to the number of elements.
If S=n; S/n=1

If S is NOT equal to n, the average can not be 1.

Let, Sum=100; n=100; S/n=100/100=1
Let, Sum=1; n=1; S/n=1/1=1
Let, Sum=10; n=9; S/n=10/9=1.(NOT EQUAL TO 1)

Sufficient. Hence the correct answer is D.

Approach Solution 3:

Any sequence of numbers has an equal spacing, with mean = median. Thus, one of the following categories of sets would be necessary for the average to be 1.

{1}
{0, 1, 2}
{-1, 0, 1, 2, 3}

In general, there should be an equal number of items that are bigger than and less than one. As you can see, there are an unusual amount of factors in this situation.

(1) If n is even, n cannot be odd, and so 1 cannot be the average. Sufficient.
(2) If S is the product of the sums of the first n numbers, then divide by n using the formula: 0 S/n 1. Since S/n (the average, or the sum of the components divided by the number of elements), we may infer that the average is not 1, but rather 1. Sufficient.

Hence the correct answer is D.

“Is the average of n consecutive integers equal to 1?” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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