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.
Seven integers, x1, x2, x3, x4, x5, x6, and x7, are picked at random from the set of all integers between 10 and 110, inclusive. If each of these integers is divided by 7 and the 7 remainders are all added together, what would be the sum of the 7 remainders?
(1) The range of the remainders is 6.
(2) The seven integers are consecutive.
There are 7 variables, so we need 7 equations in order to solve the problem; only 2 equations are given, so there is high chance (E) will be the answer.
When we look at the conditions together, the remainders become 1,2,3,4,5,6,0 so the sum becomes 1+2+3+4+5+6=21, a unique answer. The condition is sufficient, so the answer seems to be (C), but this is a question with commonly made mistakes in 4(A) (Common Mistake Type 4 4(A)).
If we look at the conditions separately,
Condition 1 gives 1,2,3,4,5,6,7==> 1+2+3+4+5+6+0=21, but this also works for
7,14,21,35,42,48,49==>0+0+0+0+0+6+0=6, so this condition is not unique, and insufficient by itself.
Condition 2, on the other hand, the sum of the remainders becomes 1+2+3+4+5+6+0=21, so this condition is unique and therefore sufficient. So the answer becomes (B).
This type of question is a common type in today GMAT math
For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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