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# If n is a positive integer, and n^2 has 25 factors, which of the follo

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Intern
Joined: 03 Sep 2016
Posts: 17
Location: India
GPA: 3.7
If n is a positive integer, and n^2 has 25 factors, which of the follo  [#permalink]

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Updated on: 30 Nov 2016, 07:47
10
00:00

Difficulty:

95% (hard)

Question Stats:

33% (02:11) correct 67% (02:00) wrong based on 100 sessions

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If n is a positive integer, and n^2 has 25 factors, which of the following must be true.

1. n has 12 factors.
2. n > 50
3. $$\sqrt n$$ is an integer.

A. 1 and 2
B. 2 only
C. 3 only
D. 2 and 3
E. none

Originally posted by Sorrow on 30 Nov 2016, 07:38.
Last edited by Bunuel on 30 Nov 2016, 07:47, edited 1 time in total.
Renamed the topic and edited the question.
Senior CR Moderator
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Re: If n is a positive integer, and n^2 has 25 factors, which of the follo  [#permalink]

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30 Nov 2016, 08:58
2
3
Sorrow wrote:
If n is a positive integer, and n^2 has 25 factors, which of the following must be true.

1. n has 12 factors.
2. n > 50
3. $$\sqrt n$$ is an integer.

A. 1 and 2
B. 2 only
C. 3 only
D. 2 and 3
E. none

$$25=1 \times 25 = 5 \times 5$$

So prime factorization of $$n^2$$ could be $$p^{24}$$ or $$p^4q^4$$ with $$p,q$$ are distinct prime numbers.
Hence, $$n$$ could be $$p^{12}$$ or $$p^2q^2$$

(1) If $$n=p^{12}$$ then $$n$$ has $$12+1=13$$ different factors, so (1) is not true.

(2) If $$n=p^2q^2$$, for example $$n=2^2 \times 3^2 =36 <50$$, so (2) is not true.

(3)
If $$n=p^{12} \implies \sqrt{n}=p^6$$ is an integer.
If $$n=p^2q^2 \implies \sqrt{n}=pq$$ is an integer.
So (3) is true.

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Re: If n is a positive integer, and n^2 has 25 factors, which of the follo  [#permalink]

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30 Mar 2018, 02:59
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If n is a positive integer, and n^2 has 25 factors, which of the follo   [#permalink] 30 Mar 2018, 02:59
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