Question

The median of a particular set of $10$ integers is itself an integer. Mark the right statement(s).
I. The integers of the set are all identical.
II. The integers of the set are consecutive.
III. The $5^{th}$ largest and $6^{th}$ largest integers of the set are odd.

• A. I only
• B. II only
• C. I and II only
• D. I and III only

Answer 1

The correct option is D I and III only

For choice I, suppose that the set is a group of ten 2s2s . The median of this set is 22 , which is an integer, so choice I could be true.

For choice II, note that for a set of ten numbers, there is no single middle number, so the median is the average of the two middle numbers (the 5th5th largest and 6th6th largest). But the average of any two consecutive integers is never an integer, so choice II can never be true. (As an example, suppose that the set is the integers from 11 to 1010 . The two middle numbers are 55 and 66 and the median is $\frac{(5+6)}{2}=\frac{5.5(5+6)}{2}=5.5$.

This reasoning helps with choice III: suppose that the middle two numbers (the 5th5th largest and 6th6th largest) are 77 and 1111 . Then the median is $\frac{(7+11)}{2}=\frac{9(7+11)}{2}=9$, which is an integer, so choice III could be true, making answer D correct.