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.
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