If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

In ascending order, let the 9 distinct integers be a, b, c, d, M, e, f, g, h, where M = median.

Statement 1: The product of the nine integers in list S is equal to the median of list S.
Thus:
a*b*c*d*M*e*f*g*h = M.
If M is a nonzero integer, we can divide each side by M, with the following result:
a*b*c*d*e*f*g*h = 1.
It is not possible that the product of 8 distinct integers is equal to 1.
Implication:
M CANNOT be a nonzero integer, implying that M=0 and that the median is NOT positive.
SUFFICIENT.

Statement 2: The sum of all nine integers in list S is equal to the median of list S.
Thus:
a+b+c+d+M+e+f+g+h = M.
Subtracting M from both sides, we get:
a+b+c+d+e+f+g+h = 0.
Implication:
As long as the sum above is equal to 0, M can be ANY INTEGER VALUE.
Case 1: -4, -3, -2, -1, 0, 1, 2, 3, 4.
Here, sum = median = 0.
Case 2: -5, -4, -3, -2, 1, 2, 3, 4, 5.
Here, sum = median = 1.
Since the median is positive in Case 2 but not in Case 1, INSUFFICIENT.

The correct answer is A.

GMATGuruNY wrote:

If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.

(2) The sum of all nine integers in list S is equal to the median of list S.

In ascending order, let the 9 distinct integers be a, b, c, d, M, e, f, g, h, where M = median.

Statement 1: The product of the nine integers in list S is equal to the median of list S.
Thus:
a*b*c*d*M*e*f*g*h = M.
If M is a nonzero integer, we can divide each side by M, with the following result:
a*b*c*d*e*f*g*h = 1.
It is not possible that the product of 8 distinct integers is equal to 1.
Implication:
M CANNOT be a nonzero integer, implying that M=0 and that the median is NOT positive.
SUFFICIENT.

Statement 2: The sum of all nine integers in list S is equal to the median of list S.
Thus:
a+b+c+d+M+e+f+g+h = M.
Subtracting M from both sides, we get:
a+b+c+d+e+f+g+h = 0.
Implication:
As long as the sum above is equal to 0, M can be ANY INTEGER VALUE.
Case 1: -4, -3, -2, -1, 0, 1, 2, 3, 4.
Here, sum = median = 0.
Case 2: -5, -4, -3, -2, 1, 2, 3, 4, 5.
Here, sum = median = 1.
Since the median is positive in Case 2 but not in Case 1, INSUFFICIENT.

The correct answer is A.

Hi Mitch,

In statement 1: how do you reduce a variable? I would do it as follows:

a*b*c*d*M*e*f*g*h = M........> M(a*b*c*d*e*f*g*h-1)=0

So either M=0 or a*b*c*d*e*f*g*h=1 (impossible) so M=0

I know that we reach same result but dividing variables could cause problems.

Am I right? or do i miss something?

Mo2men wrote:Hi Mitch,

In statement 1: how do you reduce a variable? I would do it as follows:

a*b*c*d*M*e*f*g*h = M........> M(a*b*c*d*e*f*g*h-1)=0

So either M=0 or a*b*c*d*e*f*g*h=1 (impossible) so M=0

I know that we reach same result but dividing variables could cause problems.

Am I right? or do i miss something?

Nice work!
Your solution proceeds as follows:
a*b*c*d*M*e*f*g*h = M
(a*b*c*d*M*e*f*g*h) - M = 0
M(a*b*c*d*e*f*g*h - 1) = 0.

The equation above is valid if M=0 or if a*b*c*d*e*f*g*h = 1.
As noted in my solution above, it is not possible that a*b*c*d*e*f*g*h = 1.
Thus, M=0.

In my solution, I suggested that both sides of a*b*c*d*M*e*f*g*h = M may be divided by M if M is nonzero.
Dividing both sides by M, we get:
(a*b*c*d*M*e*f*g*h)/M = M/M
a*b*c*d*e*f*g*h = 1.
The result is invalid, since it is not possible that the product of 9 distinct integers is 1.
Since M=nonzero yields an invalid result, it must be true that M=0.

GMATGuruNY wrote: In ascending order, let the 9 distinct integers be a, b, c, d, M, e, f, g, h, where M = median.

Statement 1: The product of the nine integers in list S is equal to the median of list S.
Thus:
a*b*c*d*M*e*f*g*h = M.
If M is a nonzero integer, we can divide each side by M, with the following result:
a*b*c*d*e*f*g*h = 1.
It is not possible that the product of 8 distinct integers is equal to 1.
Implication:
M CANNOT be a nonzero integer, implying that M=0 and that the median is NOT positive.
SUFFICIENT.

Hi Mitch,
For statement 1 my thought process was that because the prompt says that at least one integer in the list is negative then if we consider the following cases:
A) 1 negative integer and 8 positive integer --> Product and hence median is negative
B) 2 negative and 7 positive integer --> Product and hence the median is positive.

Could you please point out the mistake in my reasoning?

Thanks!
Manik

manik11 wrote:

GMATGuruNY wrote: In ascending order, let the 9 distinct integers be a, b, c, d, M, e, f, g, h, where M = median.

Statement 1: The product of the nine integers in list S is equal to the median of list S.
Thus:
a*b*c*d*M*e*f*g*h = M.
If M is a nonzero integer, we can divide each side by M, with the following result:
a*b*c*d*e*f*g*h = 1.
It is not possible that the product of 8 distinct integers is equal to 1.
Implication:
M CANNOT be a nonzero integer, implying that M=0 and that the median is NOT positive.
SUFFICIENT.

Hi Mitch,
For statement 1 my thought process was that because the prompt says that at least one integer in the list is negative then if we consider the following cases:
A) 1 negative integer and 8 positive integer --> Product and hence median is negative
B) 2 negative and 7 positive integer --> Product and hence the median is positive.

Could you please point out the mistake in my reasoning?

Thanks!
Manik

Your cases do not satisfy the constraint that the product of the nine distinct integers must be EQUAL to the median.
For the product to be equal to the median, THE MEDIAN MUST BE 0, as shown in my solution above.