Official Explanation:In this question, each of the two sets contains 3 consecutive positive integers and the two sets have no common numbers. We can easily imagine how these two sets could be, for example, one option would be 3, 4, 5 and 9, 10, 11.
When the two sets are combined to form a new set of 6 ordered numbers, all numbers are distinct. This is because if one number appears twice, this would mean that either the original set that the number comes from does not include consecutive integers or the two original sets have common numbers. Neither of these is possible according to the information given in this question.
We know that the median of this ordered list of 6 numbers is the average of the 3d and 4th integer which are not equal as we explained above.
This is the first hard part of this question:
How must the 3d and 4th integers relate so that their average is also an integer?
The average of two integers is also an integer if
a) the two integers are equal which is not possible according to the information given or
b) the two integers differ by an even number 2, 4, 6,...
Does b) make sense? Yes, because if you have an integer x and another integer that is say
x + 8, then their sum is always a multiple of 2 because you write it as
x + x + 8 = 2x + 8 = 2(x+4)
And thus it is divided by 2 and gives an integer. And this is the case for any even number that you can have in 8’s place above.
Also you can intuitively check b) by taking random pairs of integers that they differ by an even number. For example, 5 and 7 which differ by 2 have average equal to 6 because 5 + 7 = 12 and \(\frac{12}{2}\) = 6. Or 12 and 22 which differ by 10 have average equal to 17 because 12 + 22 = 34 and \(\frac{34}{2}\) = 17
So, the difference of the 3d and 4th integers is an even.
And how are we going to somehow relate this result to the range of the data that we finally want to find?
Here is the second tricky part of the question.
Consider the following cases:
1) If the 3d number is even then the 4th number is also an even since they differ by an even number. But since 1st, 2nd, and 3d are consecutive then 1st is also even. And since the 4th, 5th and 6th are consecutive, then the 6th number is also an even. The range is the difference of the 1st number from the 6th number and thus a difference of two even numbers which gives an even.
2) If the 3d number is odd then the 4th number is also an odd since they differ by an even number. But since 1st, 2nd, and 3d are consecutive then 1st is also odd. And since the 4th, 5th and 6th are consecutive, then the 6th number is also an odd. The range is the difference of the 1st number from the 6th number and thus a difference of two odd numbers which gives an even.
Therefore, we conclude that the range must always be an even number and the correct answers are all choices that contain even numbers
(A), (C), and (E).
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