Data Sufficiency

[GMAT math practice question]

If n is positive integer, is 21 a factor of n?

1) 21 is a factor of 3n
2) 21 is a factor of n^2

Max@Math Revolution wrote:[GMAT math practice question]

If n is positive integer, is 21 a factor of n?

1) 21 is a factor of 3n
2) 21 is a factor of n^2

Statement 1:
In other words, 3n is a multiple of 21:
3n = 21, 42, 63...
To get a list of options for n, divide every value above by 3:
n = 7, 14, 21...
If n=7, then the answer to the question stem is NO.
If n=21, then the answer to the question stem is YES.
INSUFFICIENT.

Statement 2:
Since n is a positive integer, n² must be a perfect square that is divisible by 21:
n² = 21², (2*21)², (3*21)²....
n² = 21², 42², 63²....
To get a list of options for n, take the square root of every value above:
n = 21, 42, 63...
In every case, n is divisible by 21, so the answer to the question stem is YES.
SUFFICIENT.

The correct answer is B.

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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)
If n = 21, then 21 is a factor of n and the answer is "yes".
If n = 7, then 21 is not a factor of n and the answer is "no".
Since the solution is not unique, condition 1) is not sufficient.

Condition 2)
Since 21 = 3*7 is a factor of n^2, each of 3 and 7 is a factor of n^2.
If 3 is a factor of n^2, then, since 3 is a prime number, 3 is a factor of n.
If 7 is a factor of n^2, then, since 7 is a prime number, 7 is a factor of n.
Thus, 21 is a factor of n.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

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.