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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 question asks the value of |x - y|.

Condition 1)
Since the first odd prime number is 3, we have |x - y| = 3.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
3x = 23 - 5y
If y = 1, then we have 3x = 23 - 5 = 18 or x = 6 and |x - y| = 5.
If y = 2, then we have 3x = 23 - 10 = 13 and we don't have an integer solution.
If y = 3, then we have 3x = 23 - 15 = 8 and we don't have an integer solution.
If y = 4, then we have 3x = 23 - 20 = 3 or x = 1 and |x - y| = 3
If y = 5, then we have 3x = 23 - 25 = -2 and we start to have negative numbers and we can stop this substitution process.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A