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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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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
The following reasoning shows that in the question, either p or q is an odd integer.
(p-1)(q-1) is an even integer
=> p - 1 or q - 1 is an even integer
=> p or q is an odd integer
Therefore, either p and q is an odd number, and the other one is an even number, according to condition 1. So, condition 1) is sufficient.
Condition 2)
If p is an odd number and q is an even number, then p-1 is an even number, q-1 is an odd number, and (p-1)(q-1) is an even number, which means the answer is 'yes'.
If both p and q are even numbers, then (p-1)(q-1) is an odd number, and the answer is 'no' since both p-1 and q-1 are odd numbers.
Since condition 2) does not yield a unique solution, it is not sufficient.
Therefore, A is the answer.
Answer: A
