Bunuel wrote:

Set S is comprised of six distinct positive integers less than 10. Which of the following must be true?

I: The median is an integer

II: The median is less than the average

III: The range of digits in Set S is less than 8

(A) I only

(B) I & II

(C) II & III

(D) III only

(E) None of the above.

I. The median is an integer.

Median when there are even no. of items in a set is the mean of the middle two elements.

a b [c d] e f -----> (c+d)/2

This need not be an integer all the time.

I is not an answer. Eliminate (A) and (B). III. The range of set is less than 8.

Consider the set with a 1 and a 9. For example,

1, 2, 4, 6, 7, 9.

Range = 9-1 ------> 8

III is not an answer. Eliminate (C) and (D). We can stop here. The answer by elimination is (E).

II. The median is less than average.

Consider a

equally spaced set, basically consecutive integers, where we know

mean = median.

1, 2, 3, 4, 5, 6

Median = (3 + 4)/2 ------> 7/2 -------> 3.5

Mean = (1 + 2 + 3 + 4 + 5 +6)/6 ------> 21/6 ------> 3.5 (Note you can calculate mean for the set by (1 + 6)/2, since the set is equally spaced.)

\(Mean=Median\)

This leaves (E), since none of the conditions

have to be true.