Data Sufficiency

[GMAT math practice question]

If n is a positive integer, is 91 a factor of n?

1) 91 is a factor of n^2
2) 91 is a factor of 2n

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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 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since 91 = 7*13, n2 is a multiple of both 7 and 13. Since 7 and 13 are prime numbers, n must be a multiple of both 7 and 13.
Condition 1) is sufficient.


Condition 2)
If 91 is a factor of 2n, then 91k = 2n for some integer k. Since 91 is odd, k must be an even integer. Write k = 2a, for some integer a. Then
2n = 91*2a. It follows that n = 91*a.
Thus, n is a multiple of 91.
Condition 2) is sufficient.

This is a CMT(Common Mistake Type) 4(B) question. Condition 2) is easy to understand and condition 1) is difficult to figure out. If you are unable to figure out condition 2), you should choose D as the answer.

Therefore, the answer is D.

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.

Answer: D