Is integer x odd? (1) 2x+1 is odd (2) x/2 is even

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)

Whatever integer x is, 2x+1 is an odd integer.

If x = 1, then 2x+1=3, x is an odd integer and the answer is 'yes'.
If x = 2, then 2x+1=5, x is an even integer and the answer is 'no'.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since x/2 is an even integer, we have x/2 = 2k or x = 4k for an integer k.
Then x is an even integer and the answer is 'no' only.

Since condition 2) yields a unique solution, it is sufficient.

Therefore, B is the answer.

Since 'no' is also a unique answer by CMT (Common Mistake Type) 1, condition 2) is sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

By asking us if integer x is odd, the question gives us a hidden clue that x “could be” even. We need to answer the main question with a definite YES or NO.

From statement I alone, 2x+1 = odd.

If the addition of two integers yields an even integer, then one of them HAS TO BE odd and the other HAS TO BE even. Of 2x and 1, 1 is definitely ODD and 2x is definitely EVEN. Knowing that 2x is even is not sufficient to find whether x is ODD or EVEN.

Statement I alone is insufficient to answer the question. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, \(\frac{x}{2}\) = even.
Cross multiplying, we have, x = 2*even.

The product of ANY 2 even numbers is ALWAYS even. Therefore, x is definitely even. This is sufficient to say x is NOT ODD.

Statement II alone is sufficient to answer the question with a definite NO. Answer options C and E can be eliminated.
The correct answer option is B.

In a YES-NO type of DS question, remember that a NO is as good an answer as a YES.

Hope that helps!