There are six consecutive positive integers in Set S. What

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There are six consecutive positive integers in Set S. What is the value of positive integer n?

(1) When each integer in S is divided by n, the sum of remainders is 11.
(2) When each integer in S is divided by n, the remainders include five different values.

The question states that the numbers are 6 consecutive terms, so we only need to know the first term; there is only one variable and 2 equations, so there is high chance that (D) will be the answer.
Looking at condition 1, for {1,2,3,4,5,6} and n=5, the sum of the remainder is 1+2+3+4+0+1=11, but for {2,3,4,5,6,7} and n=4, the sum of the remainder is also 2+3+0+1+2+3=11. n=4,5; no unique value of n is given, so this condition is insufficient.
Looking at condition 2, in order to have different remainders, n has to be 5. So the remainder becomes 0,1,2,3,4. This condition gives a unique value of n, so the answer becomes (B).

Normally for cases where we need 1 more equation, such as original conditions with 1 variable, or 2 variables and 1 equation, or 3 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore D has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) separately. Here, there is 59 % chance that D is the answer, while A or B has 38% chance. There is 3% chance that C or E is the answer for the case. Since D is most likely to be the answer according to DS definition, we solve the question assuming D would be our answer hence using 1) and 2) separately. Obviously there may be cases where the answer is A, B, C or E.