Kinshook wrote:
Is the square of positive integer Z divisible by 9?
(1) The sum of the digits of Z^3 is divisible by 9
(2) 3Z^4 + 16 leaves a remainder of 7 when divided by 9
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 "the square of positive integer \(Z\) is divisible by 9" is equivalent to "\(Z\) is divisible by 3".
Since we have 1 variable (\(Z\)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.
Condition 1)
The sum of the digits of Z^3 is divisible by 9 means Z^3 is divisible by 9.
It means Z itself is divisible by 3 and the answer is 'yes'.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
\(3Z^4 + 16 = 3Z^4 + 9 + 7\)
\(3Z^4\) is divisible by \(9\) in order that \(3Z^4 + 16 = 3Z^4 + 9 + 7\) has a remainder \(7\).
Then \(Z^4\) is divisible by \(3\) and \(Z\) is divisible by \(3\).
Since condition 2) yields a unique solution, it is sufficient.
Therefore, D is the answer.
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.
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