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 Post subject: Prime Numbers
PostPosted: Mon Apr 20, 2009 9:04 am 
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Joined: Sun Apr 19, 2009 7:08 pm
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I'm going through some Manhattan GMAT problems and I found a problem that I can't solve. I just don't get it. I am horrible at data sufficiency. Here is the problem:

Is the square root of x a prime number?

(1) |3x - 7| = 2x + 2
(2) x^2 = 9x

Thanks,
Kelly


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 Post subject: Re: Prime Numbers
PostPosted: Mon Apr 20, 2009 9:36 am 
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Joined: Mon Apr 06, 2009 5:44 pm
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Hey Kelly,

Thanks for the question ! Yeah data sufficiency can be intimidating for a lot of students. There are some basic strategies that you might want to employ when you encounter a data sufficiency problem. Lets briefly talk about those in hopes that they may lend some insight to the solution of the stated problem.

The data sufficiency problem is composed of three parts: a premise and two propositions. The premise is the "statement" of the problem and the two propositions are written below it. Before even considering the propositions, you should clearly understand what it is that the premise is asking. Then, you should ask yourself: what is it that I would like to see in order to solve this ?

Look at the first proposition and ask yourself if it conforms to what you are looking for or if it undermines it. Both are equally important. Do the same for the second proposition. If both are enough to solve the problem the answer is D, if neither of them work independently or together, the answer is E. If you need both, its C. And, if only one of them works its either A or B.

Following this basic outline. The square root of x, we will denote it as x^1/2. We ask ourselves, what kinds of numbers would we like to see in order for x^1/2 to be prime. Well, our prime numbers are {2, 3, 5, 7, 11, 13, 17, 19, 23, ....} their squares are {4, 9, 25, 49, 121, 169, ...} so essentially we want to know if x belongs to the latter set.

Lets look at the first proposition: |3x - 7| = 2x + 2. We solve absolute values by solving two linear equations independently:

(1) 2x + 2 = 3x - 7 ~> x = 9 and (2) 2x + 2 = 7 - 3x ~> 5x = 5 ~> x = 1


Clearly, the first proposition is insufficient. The positive square root of 9 is 3, which is prime. However the positive square root of 1 is not a prime number. So we cannot explicitly determine if x is prime or not. Lets look at the next proposition.

Lets solve for x in the equation x^2 = 9x. We can rewrite this as x^2 - 9x = x(x - 9) = 0. Clearly x is either 0 or 9. Again taking square roots of each of these will give us a prime, 3, and a non-prime 0. This proposition doesn't help us either.

Lets look at them together. The first proposed solution set is {9, 1}. The second proposed solution set is {9, 0}. From both of these, we can eliminate the 1 and the 0 since it is not common to both propositions. For the same reasons we can retain the 9. Both statements are taken to be true and both suggest 9 is a solution. However they don't both agree on 1 and 0. Since they both confirm 9. It is clear that x = 9 and that x^1/2 = 3 is a prime number. The correct answer is C.


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