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# How many positive factors does N have?

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Intern
Joined: 18 Nov 2012
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How many positive factors does N have?  [#permalink]

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26 Jan 2016, 05:55
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How many positive factors does N have?
1) N^2 has 4 positive factors
2) 2N has three positive factors
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4472
Re: How many positive factors does N have?  [#permalink]

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26 Jan 2016, 18:38
1
SamTheLearner wrote:
How many positive factors does N have?
1) N^2 has 4 positive factors
2) 2N has three positive factors

Dear SamTheLearner,
I'm happy to respond , my friend, but I believe something is awry in the question you have posted. In particular, there is absolutely no positive integer value of N that is consistent with statement #1 as you have posted it.

Any square of a positive integer has an odd number of positive factors, because all the factors can be placed in pairs except the factor that is squared. Thus, the factors of 36 are {1, 36} and {2, 18} and {3, 12} and {4, 9) and 6, for a total of nine factors altogether. If N is a positive integer, then it is impossible for it's square to have 4 positive factors. If N is not positive integer, then it doesn't make sense to talk about how many positive factors it has. General, the GMAT doesn't talk about factors at all unless the numbers involved are all integers.

Is it possible that you mistyped the question from the source. For example, this question would make mathematical sense:
How many positive factors does positive integer N have?
1) N^2 has three positive factors
2) 2N has four positive factors

As you probably recognize, any prime number has exactly two factors.

The only way a number can have three factors is when it is a square of prime number. If p is prime, the factors of p^2 are {1, p, p^2}, Statement one tells us that N is prime and thus has two factors. Statement #1 of my problem is sufficient alone and by itself.

There are two ways a number could have exactly four factors.
a) First Way: the number could be the product of two different primes. For example, 10 has four factors: {1, 2, 5, 10}.
b) Second Way: the number could the cube of a prime number. If p is prime, the factors of p^3 are {1, p, p^2, p^3}.

Thus, given my statement #2, it could be that N equals an odd prime number, so the product 2N is a product of two different prime numbers. Thus, N would be a prime number, a number with two factors. Or, it could be true that N = 4, so that 2N = 8, the cube of a prime number, which also would have four factors. In this case, N = 4, the square of a prime number has three factors. From this statement alone, we cannot give a definitive answer to the prompt question. This statement, alone and by itself, is insufficient.

My written question has an answer of (A). The original question is not solvable.

Does all this make sense?
Mike
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Mike McGarry
Magoosh Test Prep

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Re: How many positive factors does N have?  [#permalink]

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18 Mar 2019, 11:30
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Re: How many positive factors does N have?   [#permalink] 18 Mar 2019, 11:30
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