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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.
Since we have 2 variables (A and B) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2)
Condition 1) tells us that B6354A is a multiple of 9. So, A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, and A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), ... , (8,1) and (9,9).
Since condition 2) tells us that A < B, (9,9) is not possible. For all remaining pairs, the value of of A + B is 9.
Therefore, conditions 1) and 2) are sufficient, when applied together.
Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since B6354A is a multiple of 9 and A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), ... , (8,1) and (9,9).
We test these pairs individually to check whether they give rise to multiples of 99:
963540 = 11*87594 + 6 is not a multiple of 11.
863541 = 11*78503 + 8 is not a multiple of 11.
763532 = 11*69412 + 10 is not a multiple of 11.
663543 = 11*60322 + 1 is not a multiple of 11.
563544 = 11*51231 + 3 is not a multiple of 11.
463545 = 11*42140 + 5 is not a multiple of 11.
363546 = 11*33049 + 7 is not a multiple of 11.
263547 = 11*23958 + 9 is not a multiple of 11.
163548 = 11*14868 is a multiple of 11.
963549 = 11*87595 + 4 is not a multiple of 11.
(8,1) is a unique pair of (A,B).
Than we have A + B = 9.
Condition 1) is sufficient, since it yields a unique solution.
Condition 2) is obviously not sufficient.
Therefore, A is the answer.
Answer: A
