Solution:
Given: List contains n distinct integers.
To find: Are all n integers consecutive?
Analysis of statement 1: The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.I feel the best way to solve the question is by taking consecutive numbers and checking whether the given statement is true or not!!! If it satisfies the statement then answer will be “n” distinct integers are consecutive in nature else not. Let’s take set of consecutive integers as {11, 12, 13, 14}
Case 1: When the lowest number is removed: When 11 is removed, the set will be {12, 13,14},
So the average = 13.
Case 2: When the highest number is removed: When 14 is removed; the set will be {11, 12, 13}.
So the average = 12.
So, statement 1 is true for all consecutive distinct integers.
Hence statement 1 is sufficient to answer. We can eliminate options B, C and D.
Analysis of statement 2: The positive difference between any two numbers in the list is always less than n.Let’s take a set of consecutive integers as {11, 12, 13, 14} here n = 4; even if we are taking the difference between highest integer and lowest integer, i.e.
\(|14-11| = 3\). So satisfies the statement 2.
If we take this algebraically: Let the set be \({x, x+1, x+2………..x+n-1}.\)
Positive difference of highest integer and lowest integer, we get =
\(|x+n-1-x|=n-1.\)Therefore it is clear that the difference between the two numbers will be always less than n.
So, statement 2 is sufficient. We can eliminate option A.
Therefore the correct answer option is “D”.
_________________
GMAT Mentors