A set of 13 integers has a median of 10 and an average of 15.

Here's my method of finding it via concepts:
13 integers are present, and the median is 10.
In order to minimize the value of X, the largest integer, we have to maximize the value of all integers till 10.
The max value for each integer till the median is 10. So, our list has 7 integers, with value, '10'.
The sum of our list is 195.
10*number of integers till median = 10*7=70. Apart from these, we have 6 remaining integers.
Substract 70 from 195, we get 125.
So, the sum of the remaining 6 integers must be 125.
Now, in order to minimize the highest unique value, we must maximize the integers.
If we assume these to be 20, we have 6*20=120 which is less than 125.
Now assume them to be 21, we have 6*21=126, which is greater than 125.
Now write,
20 20 20 20 20 25(X) ...Matches our answer choice, but let's see if we can lessen the value of X, somehow.
20 20 20 20 21 24
20 20 20 20 22 23
20 20 20 21 21 22
After this, we will not be able to minimize the value of X, because then X would not be a UNIQUE integer in the list.
Hence, Option C is our answer.