S is a set of n consecutive positive integers. Is the mean of the set

Question

S is a set of n consecutive positive integers. Is the mean of the set a positive integer?

(1) the range of S is an even integer

(2) the median of S is a positive integer

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BrentGMATPrepNow · Answer

Given : S is a set of n consecutive positive integers

Target question :    Is the mean of the set a positive integer?  
This is a good candidate for rephrasing the target question.

There's a nice rule that says,   "In a set where the numbers are equally spaced, the mean will equal the median."  
For example, in each of the following sets, the mean and median are equal:
{7, 9, 11, 13, 15}
{-1, 4, 9, 14}
{3, 4, 5, 6}

Since we're told that S is a set of n  consecutive  positive integers, we know that the numbers in set S are equally spaced, which means  the median and mean are equal . 
So, one way to rephrase the target question as follows:
  REPHRASED target question :    Is the MEDIAN of the set a positive integer?

Aside: the video below has tips on rephrasing the target question

When I SCAN the two statements, I can immediately see that  statement 2 is SUFFICIENT , since it tells us that the median is a positive integer.

Now let's check statement 1...

Statement 1 : The range of S is an even integer  
Notice the following: 
If set S = (1, 2}, the range is 1
If set S = (8, 9, 10}, the range is 2
If set S = (4, 5, 6, 7} the range is 3
If set S = (4, 5, 6, 7, 8} the range is 4
.
.
.
As we can see, if set S has an even number of consecutive integers, the range will be odd.
Conversely, if set S has an odd number of consecutive integers, the range will be even.
 
Since statement one tells us the range is even, we can be certain that n (the number of integers) is odd. 
If set S consists of an ODD number of consecutive integers, then the median will be the middlemost value, which means the median will be an integer.
The answer to the REPHRASED target question is  YES, the median is a positive integer 
Since we can answer the  REPHRASED target question  with certainty, statement 2 is SUFFICIENT
 
Answer: D

RELATED VIDEO
 https://www.youtube.com/watch?v=MyG1K3ee69w

TestPrepUnlimited · Answer

S is an arithmetic sequence, therefore the mean = median for set S. Since the set itself consists of positive integers, we already know the median is a positive number. We only need to check whether it is an integer now (the median is either an integer + 0.5, or an integer).

Statement 1:

If the range is n, then there are n + 1 elements in the set, hence an odd number of elements in the set. Then we know the median has to be the middle element, hence the median must be an integer. Sufficient.

Statement 2:

Sufficient.

Ans: D

Bunuel · Answer

Official Magoosh Explanation:

First of all, for a set of consecutive integers, or for any set of evenly spaced numbers, the mean and the median are equal. If there's an odd number of members of the list, then the median is the middle number. If there's an even number of members of the list, then the median is the average of the two middle numbers. For example, the median of {1, 2, 3, 4, 5} is 3, a positive integer and member of the set. For consecutive integers, an even number of members would mean that the mean or median is the average of the two middle integers. For example, the median of {1,2, 3, 4} is the average of 3 and 4, that is, 3.5, not an integer. The only way the mean or median can be an integer is if the set of consecutive integers has an odd number of members.

Statement #1: If there are an even number of consecutive integers, then the evens and odds are balanced in the set, and the first and last number must be opposite: one must be even and the other must be odd. Thus, the range, the difference of (max) – (min) would be either (even) – (odd) or (odd) – (even), in either case, an odd number. If the range is odd, the number of consecutive integers is even.

If there are an odd number of consecutive integers, then the first and last numbers are either both even or both odd. The range would be either (even) – (even) or (odd) – (odd), in either case, an even number. If the range is even, the number of consecutive integers is odd. That must be the case here. As we have seen above, this means the mean or median is a positive integer. This statement, alone and by itself, is sufficient.

Statement #2: As we discussed above, the mean = the median. If the latter is a positive integer, so is the former. This statement, alone and by itself, is sufficient.

Both statement are separately sufficient. Answer = (D)

Victor19 · Answer

For a set of positive integers, the mean definitely has to be positive. Is the question right or am I missing something here ?

Bunuel · Answer

The mean of a set of n consecutive positive integers will for sure be positive but not necessarily an integer. The mean of a set of n consecutive integers is integer when n is odd, when n is even the mean will be integer/2. For example:

{1, 2, 3} --> mean = 2 = integer;
{1, 2, 3, 4} --> mean = 5/2 = not an integer.

So, the question basically asks whether n is odd.

Victor19 · Answer

Bunuel,

Thanks for the response. I certainly didn't pay much attention to the "integer" part which I should have. Now it is clear.

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S is a set of n consecutive positive integers. Is the mean of the set

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S is a set of n consecutive positive integers. Is the mean of the set

Prep Toolkit

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