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

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

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New post Updated on: 30 Nov 2016, 08:47
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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, 08:38.
Last edited by Bunuel on 30 Nov 2016, 08:47, edited 1 time in total.
Renamed the topic and edited the question.
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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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New post 30 Nov 2016, 09:58
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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.

The answer is C
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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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New post 30 Mar 2018, 03:59
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Re: If n is a positive integer, and n^2 has 25 factors, which of the follo &nbs [#permalink] 30 Mar 2018, 03:59
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