abhirup1711 wrote:T is a finite set of consecutive odd integers and contains atleast 3 numbers.Is the median of T an odd integer?
1.If the smallest and largest numbers are removed from T, the resulting median of the set is an odd integer.
2.If the smallest and largest numbers are removed from T, the range of the resulting set is divisible by 4.
CASE 1: Set T contains an ODD number of integers.
Here, when the largest and smallest numbers are removed, each statement is satisfied:
{1, 3, 5} --> {3}
Resulting median = 3
Resulting range = 3-3 = 0.
Since the resulting median is odd and the resulting range is a multiple of 4, each statement is satisfied.
(Note that 0 is a multiple of every positive integer.)
[3, 5, 7, 9, 11} --> [5, 7, 9}
Resulting media = 7.
Resulting range = 9-5 = 4.
Since the resulting median is odd and the resulting range is a multiple of 4, each statement is satisfied.
{9, 11, 13, 15, 17, 19, 21} --> {11, 13, 15, 17, 19}
Resulting median = 15.
Resulting range = 19-11 = 8.
Since the resulting median is odd and the resulting range is a multiple of 4, each statement is satisfied.
CASE 2: Set T contains an EVEN number of integers.
Here, when the largest and smallest numbers are removed, neither statement is satisfied.
{1, 3, 5, 7} --> {3, 5}
Resulting median = (3+5)/2 = 4.
Resulting range = 5-3 = 2.
Since the resulting median is even and the resulting range is not a multiple of 4, neither statement is satisfied.
[3, 5, 7, 9, 11, 13} --> [5, 7, 9, 11}
Resulting median = (7+9)/2 = 8.
Resulting range = 11-5 = 6.
Since the resulting median is even and the resulting range is not a multiple of 4, neither statement is satisfied.
{9, 11, 13, 15, 17, 19, 21, 23} --> {11, 13, 15, 17, 19, 21}
Resulting median = (15+17)/2 = 16.
Resulting range = 21-11 = 10.
Since the resulting median is even and the resulting range is not a multiple of 4, neither statement is satisfied.
Thus:
To satisfy statement 1, set T must be composed of an odd number of consecutive odd integers.
To satisfy statement 2, set T must be composed of an odd number of consecutive odd integers.
The median of an odd number of consecutive odd integers will always be odd.
Thus, each statement by itself is SUFFICIENT.
The correct answer is
D.
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