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.

If x and y are positive integers, what is the remainder when xy is divided by 4?

(1) x + y = xy

(2) (x – 1)(y – 1) is odd.

There are 2 variables and 2 equations are given from the 2 conditions, so there is high chance (C) will be our answer.

From condition, we only get x=y=2 as an answer, because xy-x-y=0, (x-1)(y-1)=1, x-1=1, y-1=1즉 x=y=2. This is sufficient.

From condition 2, in order for (x-1)(y-1)=odd, x-1=odd, y-1=odd, and x=1+odd=even, y=1+odd=even. Therefore, xy=even*even=4m(a multiple of 4), and the remainder when this is divided by 4 always becomes 0, so this is sufficient.

The answer becomes (D) according to common mistake type 4(B).

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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