Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.
Is positive integer n – 1 a multiple of 3?
(1) n^3 – n is a multiple of 3
(2) n^3 + 2n^2+ n is a multiple of 3
When you modify the original condition and the question, they become n-1=3t(t is a positive integer)? --> n=3t+1?. There is 1 variable(n), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), it becomes n^3-n=(n-1)n(n+1). The multiplication of three consecutive integers is always a multiple of 6. So, n=3 -> no, n=4 -> yes, which is not sufficient.
For 2), n^3 + 2n^2+ n=3k(k is a positive integer) → n(n+1)^2=3k. In n(n+1)^2=3k, either n=3k or n=3k-1 should be valid. So, it is always no and sufficient.
Therefore, the answer is B.
--> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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