vrn2vw wrote:The positive integer k has exactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, including 1 and k, what is the value of K?

(1) 3^2 is a factor of k
(2) 7^2 is NOT a factor of k

May I receive some help with this problem? It's from the official GMAT Prep application.


OA is D

From the stem, we know that K's factors are 1, 3, 7, 21 (3*7), __, and K.

Statement 1: This tells us there are two factors of 3, so 9 is also a factor of K. K's factors are 1, 3, 7, 9, 21, and K. Since there are two 3's and a 7 in K's factors, then 3*3*7 = 63 is also a factor.

Therefore K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

Statement 2: If there are not 2 7's in K's factors, and there are exactly 6 factors total, there must be two factors of 3. Otherwise, if we were to use a non-prime factor, then K would have more than 6 factors. (Remember 'K' has exactly two positive prime factors)

Therefore, K's factors are 1, 3, 7, 9, 21, 63.
SUFFICIENT

Answer: Option D

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.

The positive integer k has eactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, incuding 1 and k, what is the vlaue of k?
1) 3^2 is a factor of k
2) 7^2 is not a factor of k

We get k=(3^2)7 or (7^2)3 so that the number of factors becomes 6[=(2+1)(1+1)]
There is one variable (k) and 2 equations are given by the 2 conditions, so there is high chance (D) will be the answer.
For condition 1, we get k=(3^2)7 , an unique answer making the condition sufficient.
For condition 2, we get k=(3^2)7, an unique answer making the condition sufficient as well
Therefore the answer becomes (D)

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.