Ten different integers are between 1 and 100, inclusive. if the median of the ten integers is m, which of the following must be true ?
(A) m is not an integer.
(B) m is even.
© m is odd.
(D) m ≥ 6.
(E) m ≤ 96
*Please explain.
Recall that if a set S has an even # of elements, the median is defined as the average of the middle two (after S is sorted).
A, B, and C are not true because we can easily pick two integers in the middle such that their average (median) is an integer, or odd or even.
D is not true because we can let S = {1,2,3,4,5,6,…} and the median is 5.5.
E is true because the largest possible value of m occurs when S = {…, 95,96,97,98,99,100} and m = 95.5 < 96 in this case.