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A challenging problem. Not sure if I could explain clearly.

I. does not give much detail. The numbers can be any where from x-2 to x+2.

II. If the average and mode are the same (x) and when the numbers arrainged ascending, x has to be in the middle (median).
x is the most occuring number (since mode = x) and if this occurs on either side of the middle, the averagewill be tilted to that side.
Hence, median should also be x. And median-mode = x-x = 0

1) It is not sufficient. Let us assume the following sets

1,2,3,4,5 and 1,3,5,7,9

Both fulfill the requirement that the difference between any 2 integer is less than 3, yet we cannot determine exactly what is difference between the median and mode.

Statement 1 tells us that the difference between any two integers in the set is less than 3. This information alone yields a variety of possible sets.

For example, one possible set (in which the difference between any two integers is less than 3) might be:

(x, x, x, x + 1, x + 1, x + 2, x + 2)

Mode = x (as stated in question stem)
Median = x + 1
Difference between median and mode = 1

Alternately, another set (in which the difference between any two integers is less than 3) might look like this:

(x â€“ 1, x, x, x + 1)

Mode = x (as stated in the question stem)
Median = x
Difference between median and mode = 0

We can see that statement (1) is not sufficient to determine the difference between the median and the mode.

Statement (2) tells us that the average of the set of integers is x. This information alone also yields a variety of possible sets.

For example, one possible set (with an average of x) might be:

(x â€“ 10, x, x, x + 1, x + 2, x + 3, x + 4)

Mode = x (as stated in the question stem)
Median = x + 1
Difference between median and mode = 1

Alternately, another set (with an average of x) might look like this:

(x â€“ 90, x, x, x + 15, x + 20, x + 25, x + 30)

Mode = x (as stated in the question stem)
Median = x + 15
Difference between median and mode = 15

We can see that statement (2) is not sufficient to determine the difference between the median and the mode.

Both statements taken together imply that the only possible members of the set are x â€“ 1, x, and x + 1 (from the fact that the difference between any two integers in the set is less than 3) and that every x â€“ 1 will be balanced by an x + 1 (from the fact that the average of the set is x). Thus, x will lie in the middle of any such set and therefore x will be the median of any such set.

If x is the mode and x is also the median, the difference between these two measures will always be 0.

The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Perfect. Yes, it should be C.
I think I made a mistake by overseeing the possibility that the average tilt towards a particular side can be levelled by using a huge number on the other side (like x-10 and x-90 in your examples).

It wouldn't have been clear, if not for your long explanation. Thanks.