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Are all n integers consecutive?

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Are all n integers consecutive?

by rsarashi » Sun Jun 11, 2017 6:46 am
A list contains n distinct integers. Are all n integers consecutive?

(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.

(2) The positive difference between any two numbers in the list is always less than n.

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by [email protected] » Sun Jun 11, 2017 12:00 pm
Hi rsarashi,

This DS question has a complex 'feel' to it, so you might find it useful to deal with the information in a slightly different "order"

We're told that we have N DISTINCT integers (meaning that the numbers are all different - NO duplicates). We're asked if the N integers are consecutive. This is a YES/NO question.

To start, if we ARE dealing with consecutive integers, then there will be a variety of built-in patterns that will define the integers as consecutive (and if we can spot any of those patterns, then we might be able to reverse-engineer that the group is comprised of consecutive integers). Fact 2 is the easier of the two options, so I'm going to start there...

2) The positive difference between ANY two numbers in the list is always LESS than N.

IF.... N = 2, then the difference between those two integers has to be LESS than 2. Since we already know that the numbers are DISTINCT, the two values would have to have a difference of 1. For example, the list could be {0,1}, {3,4}, {-9, -8}, etc. This means that they ARE consecutive integers and the answer to the question is YES.

IF... N = 3, then the difference between any two of those three integers has to be LESS than 3. Since we already know that the numbers are DISTINCT, then any two values would have to have a difference of 1 or 2. The group is 3 integers though, so again - they will have to be consecutive. For example {1, 2, 3}, {7, 8, 9}, {-2, -1, 0}, etc. The answer to the question is also YES.

Increasing the number of integers will NOT change the outcome (try an example with 4 or 5 integers and you'll see). The answer to the question will ALWAYS be YES.
Fact 2 is SUFFICIENT.

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.

Since Fact 2 found a 'quirky' way to define that the list of integers was consecutive, it gets me thinking that maybe Fact 1 has also done that. However, we need proof before we make our assessment, so let's start off with a consecutive list of integers and see if it 'fits' what Fact 1 describes.

IF... we're dealing with {1, 2, 3}, then N = 3.
Remove the lowest value and the average becomes (2+3)/2 = 2.5
Remove the largest value and the average becomes (1+2)/2 = 1.5
The difference in averages IS exactly 1, so this example 'fits' what we're told and the answer to the question is YES. (incidentally, if you try any other group of consecutive integers, you'll find that this pattern holds true - you should try a few and prove it for yourself).

Now let's try a group of numbers that is NOT consecutive....
IF... we're dealing with {1, 2, 4}, then N = 3.
Remove the lowest value and the average becomes (2+4)/2 = 3
Remove the largest value and the average becomes (1+2)/2 = 1.5
The difference here is NOT 1 though, so this does NOT 'fit' what we were told and is NOT a valid example. Looking at this example, it seems that if the largest number is "too far" from the smallest number, and there are 'missing' integers in between, then the difference in averages will NOT equal 1 (it will be LARGER every time). With a few additional examples, you can prove it.

Thus, the only groups of numbers that will fit Fact 1 are consecutive integers and the answer to the question will always be YES.
Fact 1 is SUFFICIENT.

Final Answer: D

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by Jay@ManhattanReview » Tue Jun 13, 2017 4:21 am
rsarashi wrote:A list contains n distinct integers. Are all n integers consecutive?

(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.

(2) The positive difference between any two numbers in the list is always less than n.

OAD
Given: There are n distinct integers.

Question: Are these all n integers consecutive?

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.

Case 1: Say the number of integers is odd and consecutive: n, (n+1), and (n+2).

Current mean: [n + (n+1) + (n+2)]/3 = n + 1
Mean after the removal of the smallest number, n = [(n+1) + (n+2)]/2 = n + 1.5 ----(1)
Mean after the removal of the largest number, n = [n + (n+1)]/2 = n + 0.5 ----(2)

We see that (1) = 1 + (2)
Case 1 is a valid scenerio.

Case 2: Say the number of integers is even and consecutive: n, (n+1), (n+2),and (n+3).

Current mean: [n + (n+1) + (n+2) + (n+3)]/4 = n + 1.5
Mean after the removal of the smallest number, n = [(n+1) + (n+2) + (n+3)]/3 = n + 2 ----(1)
Mean after the removal of the largest number, n = [n + (n+1) + (n+2)]/3 = n + 1 ----(2)

We see that (1) = 1 + (2)
Case 2 also a valid scenerio.

Case 3: Say the number of integers is odd and non-consecutive: n, (n+2), and (n+4).

Current mean: [n + (n+2) + (n+4)]/3 = n + 2
Mean after the removal of the smallest number, n = [(n+2) + (n+4)]/2 = n + 3 ----(1)
Mean after the removal of the largest number, n = [n + (n+2)]/2 = n + 1 ----(2)

We see that (1) ≠ 1 + (2)
Case 3 is not a valid scenario. Thus, we cannot have a scenario such that 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. Thus, Statement 1 itself is sufficient.

Statement 2: The positive difference between any two numbers in the list is always less than n.

Since the integers are distinct, this fact is possible only if the n integers are distinct and consecutive.

Case 1: n = 2

There are two integers; since the positive difference between them must be less than 2, it can only by 0 or 1. The difference cannot be 0 as the integers are distinct, thus, the positive difference between the integers is 1, implying they are consecutive.

Case 2: n = 3

There are three integers; since the positive difference between ANY OF THEM must be less than 3, it can only by 0, 1, or 2. The difference cannot be 0 as the integers are distinct, thus, the positive difference between the integers is 1 or 2. The difference between the first integer and the third integer cannot be 1 since 1 is the difference between the first and the second integer, implying that all n are consecutive integers. Sufficient.

The correct answer: E

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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