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17 Mar 2015, 05:58
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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(hard)
Question Stats:
27% (02:52) correct
73%
(02:29)
wrong
based on 673
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The numbers D, N, and P are positive integers, such that D < N, and N is not a power of D. Is D a prime number?
(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2
(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2
23 Mar 2015, 05:57
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Bunuel
MAGOOSH OFFICIAL SOLUTION:
This is a tricky one.
Statement #1: two kinds of numbers have exactly four factors: (a) products of two distinct prime numbers, and (b) cubes of prime numbers.
The product of two distinct prime numbers S and T would have factors {1, S, T, ST}. For example, the factors of 10 are (1, 2, 5, 10), and the factors of 21 are {1, 3, 7, 21}.
The cube of a prime number S would have as factors 1, S, S squared, and S cubed. For example, 8 has factors {1, 2, 4, 8} and 27 has factors {1, 3, 9, 27}.
We know N is not a power of D, so the second case is excluded. N must be the product of two distinct prime numbers. We know D < N, so of the four factors, D can’t be the product of the two prime numbers. D could be either of the prime number factors, or D could be 1, which is not a prime number. Because D could either be a prime number or 1, we cannot give a definitive answer to the question. This statement, alone and by itself, is not sufficient.
Statement #2: this is tricky. The first few plug-ins seem to reveal a pattern.
Even if you sense a pattern, it’s important to remember that plugging in numbers alone is never enough to establish that a DS statement is sufficient. Here, if we persevered to one more plug-in, we would find the one that breaks the pattern.
P = 5 --> 3^5 + 2 = 243, which is not a prime.
That gives another answer to the prompt, so we know this statement is not sufficient.
To avoid a lot of plugging in, it’s also very good to know that in mathematics, prime numbers are notorious for not following any easy pattern. It is impossible to produce an algebraic formula that will always produce prime numbers. In fact, this is more than you need to know, but the hardest unsolved question in higher mathematics, the Riemann Hypothesis, concerns the pattern of prime numbers; mathematicians have been working on this since 1859, and no one has proven it yet. Suffice to say that no one-line algebraic formula is going to unlock the mystery of prime numbers!
Combined statements: according to the information in statement #1, either D = 1 or D is a prime number. Well, statement #2 excludes the possibility that D = 1, because that number cannot be written as two more than a power of 3. Therefore, D must be a prime number. We have a definitive answer to the prompt question. Combined, the statements are sufficient.
Answer = (C)
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17 Mar 2015, 07:51
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I will say C. Please let me know if I am missing something.
(1) The four factors of N include 1, D, something, and N. Therefore, N is the product of D and something.
Test case 1: N = 6 so D is either 2 or 3 and therefore is prime.
Test case 2: N = 8 so D is 4 and not prime (D cannot be 2 because 8 is a power of 2, which violates the constraints.)
(2) Possible values for D include 5, 11, 29, 83, 245, ... etc.
We have prime and non-prime possibilities, so this is insufficient.
Combining the statements, D must be one of the prime values given in statement 2, otherwise N would have more than four factors.
(1) The four factors of N include 1, D, something, and N. Therefore, N is the product of D and something.
Test case 1: N = 6 so D is either 2 or 3 and therefore is prime.
Test case 2: N = 8 so D is 4 and not prime (D cannot be 2 because 8 is a power of 2, which violates the constraints.)
(2) Possible values for D include 5, 11, 29, 83, 245, ... etc.
We have prime and non-prime possibilities, so this is insufficient.
Combining the statements, D must be one of the prime values given in statement 2, otherwise N would have more than four factors.
