byRituparna Nath Content Writer at Study Abroad Exams
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
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are not sufficient.
‘S is a set of n consecutive positive integers. Is the mean of the set a positive integer?’ - is a topic that belongs to the GMAT Quantitative reasoning section of GMAT. The question is borrowed from the book “GMAT Quantitative Review”. The GMAT Quant section mainly comprises a set of 31 questions that need to be answered logically. GMAT Data Sufficiency questions include a problem statement that is heeded by two factual arguments. GMAT data sufficiency consists of 15 questions which are two-fifths of the entire sum of 31 GMAT quant questions.
Solution and Explanation
Approach Solution 1:
The mean and the median are equivalent for any group of consecutive integers or any group of numbers with equal spacing. The median is the middle integer if the list has an odd number of members. The median is the average of the two middle numbers when the number of list members is even. As an illustration, the set's median for the numbers 1, 2, 3, and 4 is 3, a positive integer. When there are two consecutive integers, the mean or median is the average of the two middle integers if there are an even number of members.
Example- The median of 1, 2, 3, and 4 is 3.5, not 1, 2, or any other integer, which is the average of 3 and 4. If the sequence of consecutive numbers includes an odd number of members, then the mean or median can only be an integer.
Now looking at statement 1 - the range of S is an even integer
The evens and odd in the set are balanced if there are an even number of consecutive integers, and the first and last numbers must be the opposite: one must be even and the other must be odd. As a result, the range, or difference between (max) and (min), would be either (even) - (odd) or (odd) - (even), which is always an odd number. If the range is odd, then there are even more consecutive integers.
The first and last numbers are either both even or both odd if there are an odd number of consecutive integers. The range would be an even number, either (even) - (even) or (odd) - (odd). The number of successive integers is odd if the range is even. This must be the situation. This indicates that the mean or median is a positive integer, as we have seen before. This assertion stands alone and is sufficient.
Now looking at statement 2 - the median of S is a positive integer
The mean = the median, as was previously mentioned. The former is also a positive integer if the latter is. This assertion stands alone and is sufficient.
Therefore, statement 1 and statement 2 are sufficient
Answer is D, which is statement 1 and statement 2 are sufficient.
Correct Answer: D
Approach Solution 2:
There is another approach to this question which is fairly simple
Let us rephrase the question
The mean will equal the median in a collection where the numbers are evenly spaced apart, according to a rule.
For instance, the median and mean in each of the sets below are equal.
{7, 9, 11, 13, 15}
{-1, 4, 9, 14}
{3, 4, 5, 6}
We know that the numbers in set S are evenly spaced because we are told that it is a collection of n successive positive integers. As a result, we may infer that the median and mean are both equal.
Thus, the following is one way to rephrase the target question:
Is the MEDIAN of the set a positive integer?
Looking at statement 2 - the median of S is a positive integer.
The fact that statement 2 informs us that the median is a positive integer allows us to immediately determine that it is sufficient
Looking at statement 1 - The range of S is an even integer
Let us evaluate
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, the range will be odd if set S contains an even number of consecutive integers.
In contrast, the range will be even if set S contains an odd number of consecutive integers.
We can be sure that n (the number of integers) is odd because assertion one informs us that the range is even.
The median will be the middle value, which implies that it will be an integer, if set S consists of an odd number of successive numbers.
The median is a positive integer, in response to the REPHRASED target question.
Theredoire, statement 1 is sufficient
Answer is D, which is statement 1 and statement 2 are sufficient.
Correct Answer: D
Approach Solution 3:
It is given that S is an arithmetic sequence, therefore the mean = median for set S. The set itself consists of positive integers, we already know a median is a positive number. The question asks to check whether it is an integer and whether 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. So, an odd number of elements in the set. The median has to be the middle element, hence the median must be an integer. Sufficient.
Statement 2: Sufficient.
Correct Answer: D
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