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Math Expert V
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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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Difficulty:   95% (hard)

Question Stats: 28% (01:53) correct 72% (01:52) wrong based on 74 sessions

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

I. n has 12 factors.
II. n > 50
III. $$\sqrt n$$ is an integer.

A. I and II only
B. II only
C. III only
D. II and III
E. None

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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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for given condition
n^2 has 25 factors
IMO C III is valid

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

I. n has 12 factors.
II. n > 50
III. $$\sqrt n$$ is an integer.

A. I and II only
B. II only
C. III only
D. II and III
E. None

Are You Up For the Challenge: 700 Level Questions
Intern  B
Joined: 17 Mar 2018
Posts: 6
Re: If n is a positive integer, and n^2 has 25 factors, which of the follo  [#permalink]

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Why is n not greater than 50? Can someone explain?
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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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In this case, we can create a number with 25 factors.
If the prime factorization of x is x =(2^a)*(3^b)*(5^c) .... the number of factors is (a+1)(b+1)(c+1) ...
For example, 48 = 16 * 3 = $$2^4 * 3$$ has $$(4+1)*(1+1) = 5 * 2 = 10$$ factors.

So in this case, the number of factors, 25, must be broken down into 5 * 5 = 25 or 25 * 1 = 25. Using 5 * 5 = 25, we can have (a + 1)(b + 1) = 25, a = 4 and b = 4. This means $$x = 2^4*3^4 = n^2$$. Thus $$n = 2^2*3^2$$ which has 3*3 = 9 factors. We may check for the other case, $$n^2 = 2^{24}$$ -> $$n = 2^{12}$$ gives 13 factors). In both cases n has an odd number of factors so it is a square, hence III is true. Finally, we had $$n = 2^2*3^2 = 36$$ so II is false. (Note all of these n's are just minimum values of n, in fact n can be (any prime number)^4 * (any other prime number)^4 etc)

Ans: C

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

I. n has 12 factors.
II. n > 50
III. $$\sqrt n$$ is an integer.

A. I and II only
B. II only
C. III only
D. II and III
E. None

Are You Up For the Challenge: 700 Level Questions

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Originally posted by TestPrepUnlimited on 11 Nov 2019, 05:58.
Last edited by TestPrepUnlimited on 13 Nov 2019, 09:50, edited 1 time in total.
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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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Bunuel wrote:
If n is a positive integer, and n^2 has 25 factors, which of the following must be true.

I. n has 12 factors.
II. n > 50
III. $$\sqrt n$$ is an integer.

A. I and II only
B. II only
C. III only
D. II and III
E. None

Are You Up For the Challenge: 700 Level Questions

Statement I. Let n = p^r where p is a prime. Then n^2 = p^(2r). Since n^2 has 25 factors, 2r + 1 = 25 or r = 12. However, if n = p^12, then n has 12 + 1 = 13 factors. We see that I is not necessarily true.

Statement II. If n = 36 = 2^2 x 3^2, then n^2 = 2^4 x 3^4 has (4 + 1) x (4 + 1) = 25 factors. We see that n < 50. So II is not necessarily true either.

Statement III. From the above, we see that the only way n^2 can have 25 factors is when n = p^12 or n = p^2 x q^2 where p and q are primes. In either case, we see that n is a perfect square (notice that in the former case, n = (p^6)^2 and in the latter case, n = (p x q)^2). Therefore, √n is an integer. Statement III must be true.

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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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aashraiarun wrote:
Why is n not greater than 50? Can someone explain?

aashraiarun

n can be (2^2) (3^2) = 36

((2^2) (3^2))^2 = 25 factors If n is a positive integer, and n^2 has 25 factors, which of the follo   [#permalink] 20 Nov 2019, 12:21
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