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# S97-01

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Math Expert
Joined: 02 Sep 2009
Posts: 58383

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16 Sep 2014, 01:51
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Difficulty:

55% (hard)

Question Stats:

58% (00:59) correct 42% (00:54) wrong based on 65 sessions

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How many factors does $$x$$ have, if $$x$$ is a positive integer?

(1) $$x = p^n$$, where $$p$$ is a prime number

(2) $$n^n = n + n$$, where $$n$$ is a positive integer

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16 Sep 2014, 01:51
Official Solution:

We cannot easily rephrase the question. Note that we may not need to know $$x$$ in order to know how many factors it has.

Statement (1): INSUFFICIENT. Without knowing the value of $$n$$, we cannot determine the number of factors $$x$$ has.

Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable $$n$$, whereas the question only involves the variable $$x$$.

Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of $$n$$.

Since $$n$$ is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.

$$1^1 = 1 + 1$$? No.

$$2^2 = 2 + 2$$? Yes.

$$3^3 = 3 + 3$$? No.

$$4^4 = 4 + 4$$? No.

Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of $$n$$. Thus, we can determine that the value of $$n$$ is 2.

Now, we do not know the value of $$p$$, nor of $$x$$, but we do now know that $$x = p^2$$, with $$p$$ as a prime number. Since a prime number has no factors other than 1 and itself, we can see that $$x$$ has no factors other than 1, $$p$$, and $$p^2$$. Thus, $$x$$ has exactly 3 factors, and we can answer the question definitively.

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Joined: 23 May 2017
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03 Jul 2017, 03:25
Combining statements 1 and 2, for all other prime numbers other than 1, the factors are 1, p, and p2. what if the prime number is 1, it has only 1 factor. so how is it definitive ? I would say both statements together are insufficient. Can you please explain ?

thanks
Math Expert
Joined: 02 Sep 2009
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03 Jul 2017, 03:29
Ganeshsrinivasan wrote:
Combining statements 1 and 2, for all other prime numbers other than 1, the factors are 1, p, and p2. what if the prime number is 1, it has only 1 factor. so how is it definitive ? I would say both statements together are insufficient. Can you please explain ?

thanks

1 is NOT a prime number.

A Prime number is a positive integer with exactly two distinct positive divisors: 1 and itself. The smallest prime (and the only even prime) is therefore 2.

For more on Number Theory check the following post: https://gmatclub.com/forum/math-number- ... 88376.html
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01 Feb 2019, 00:12
What about 1 for a prime number? that only has 1 factor so you can not tell if it will have 3 or 1 factors. Not sufficient
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01 Feb 2019, 00:14
jtwill5 wrote:
What about 1 for a prime number? that only has 1 factor so you can not tell if it will have 3 or 1 factors. Not sufficient

Please check the post just above yours: https://gmatclub.com/forum/s97-184689.html#p1880964
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Re: S97-01   [#permalink] 01 Feb 2019, 00:14
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# S97-01

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