sukhman wrote:Given a series of n consecutive positive integers, where n > 1, is the average value of this series an integer divisible by 3?
(1) n is odd
(2) The sum of the first number of the series and (n - 1) / 2 is an integer divisible by 3
Dear
sukman,
This is an interesting question. I am happy to help.
First of all, if we have an even number of consecutive integers, e.g. {1, 2, 3, 4, 5, 6}, the average is
not an integer --- the average of that list is 4.5. To get an integer average, the number of consecutive integers must be odd. If the number is odd, e.g. {1, 2, 3, 4, 5, 6, 7}, the average is the middle number, here 4.
Statement #1:
n is odd
Well, the average will be an integer, the middle integer on the list, but this middle number may or may not be divisible by 3.
Set = {2, 3, 4} --- prompt answer = "yes"
Set = {3, 4, 5} --- prompt answer = "no"
We can construct different scenarios consistent with this statement that produce different answers to the prompt, so this statement, alone and by itself, is
not sufficient.
Statement #2:
The sum of the first number of the series and (n - 1)/2 is an integer divisible by 3
If n is even, then (n - 1)/2 isn't an integer at all, so even values of n would not even be consistent with this statement. This statement implies the true of statement #1, which is the sign of poorly written DS question.
Suppose n is odd. Here's one set for which this statement is true:
{2, 3, 4}, so the first number is 2, n = 3, so (3 - 1)/2 = 1, and the sum required is 3, which is divisible by 3. As it happens, this set produces a "yes" answer to the prompt question.
Here's another set:
{7, 8, 9, 10, 11}, so the first number is 7, n = 5, so (5 - 1)/2 = 2, and 7 + 2 = 9, so this also produces a "yes" answer to the prompt question.
In fact, if a is the first number, (n - 1)/2 is half the set, and a + (n - 1)/2 will always be the middle number of the set. If the middle number of the set is divisible by 3, then the average of the set is divisible by three, because the middle number is the average. Thus, this statement, by itself, is sufficient, but this is awkward, because this statement directly implies statement #1.
Thus, the correct answer for this question is not well-defined. I would say [spoiler]
(B)[/spoiler], but I could see that someone could argue [spoiler]
(C)[/spoiler] legitimately. Basically, this question appears to be written by someone who understand math well but who doesn't really understand the logic of GMAT DS well. This question falls quite short of the high standards of the GMAT.
Does all this make sense?
Mike
