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# If Q, a positive integer, has 5 factors, which of the foll

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Joined: 09 Feb 2013
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If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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Updated on: 12 Mar 2013, 02:14
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64% (00:54) correct 36% (00:56) wrong based on 299 sessions

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If Q, a positive integer, has 5 factors, which of the following must be true about Q?

I. Q is the square of a prime number.
II. Q is the fourth power of a prime number.
III. Q is the product of two prime numbers.

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

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Originally posted by emmak on 12 Mar 2013, 00:33.
Last edited by Bunuel on 12 Mar 2013, 02:14, edited 1 time in total.
Edited the question.
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Joined: 02 Sep 2009
Posts: 46207
Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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12 Mar 2013, 02:22
emmak wrote:
If Q, a positive integer, has 5 factors, which of the following must be true about Q?

I. Q is the square of a prime number.
II. Q is the fourth power of a prime number.
III. Q is the product of two prime numbers.

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Q to have 5 factors it should be of the form Q = prime^4 (the number of factors 4+1=5), so II is always true (C and D remains) and I is never true (only C remains).

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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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12 Mar 2013, 02:42
emmak wrote:
If Q, a positive integer, has 5 factors, which of the following must be true about Q?

I. Q is the square of a prime number.
II. Q is the fourth power of a prime number.
III. Q is the product of two prime numbers.

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

If Q has 5 factors, we can represent Q = a$$^4$$, where a is positive integer more than 1.Let's assume that "a" is not a prime number. Let a = kp, where both k and p are positive integers.

Thus, Q = $$(kp)^4 = k^4*p^4$$. Now the number of factors of Q = (4+1)*(4+1) = 25. But as the given condition states that Q has ONLY 5 factors, thus "a" can't have any other factor except 1 and itself. Thus, a = prime number.

Statement I :We can represent Q = (a^2)^2. Thus, we have to prove whether a^2 is a prime number. Take a=2. We can see that it is not a prime number. Thus, this option can't answer a "MUST be true question"

Statement II : Always true as proved above.

Statement III : Again take a =2. Thus, Q = 64. We don't have this as product of 2 primes.

B.
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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17 Mar 2013, 10:14
Hi Bunnuel, can you explain that a positive number with 5 factors should be of the form Q = prime^4 ?? If i have a number with factors such as 2,3,4 the the positive number comes out to be 24 which has 5 factors 1, 2, 3, 4, and 24 itself. Now can you apply the concept of Q = prime^4 to it please ??
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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17 Mar 2013, 10:50
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1
tanveer123 wrote:
Hi Bunnuel, can you explain that a positive number with 5 factors should be of the form Q = prime^4 ?? If i have a number with factors such as 2,3,4 the the positive number comes out to be 24 which has 5 factors 1, 2, 3, 4, and 24 itself. Now can you apply the concept of Q = prime^4 to it please ??

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24, so 8 factors.

Below might helps to understand the answer.

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

According to the above, if and only Q=prime^4 it will have (4+1)=5 factors. For example, Q=2^4=16 --> 16 have the following factors: 1, 2, 4, 8, and 16.

Similar questions to practice:
how-many-odd-positive-divisors-does-540-have-106082.html
how-many-factors-does-36-2-have-126422.html
how-many-different-positive-integers-are-factor-of-130628.html
how-many-distinct-positive-factors-does-30-030-have-144326.html
how-many-distinct-integers-are-factors-of-147038.html

Hope it helps.
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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17 Mar 2013, 11:44
Woww !!! Thanks a lot Bunuel !! This was my first post and i got such a detailed and kind reply..Thank you and thank you gmatclub!! Thanks Walker !!
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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18 Mar 2013, 02:34
emmak wrote:
If Q, a positive integer, has 5 factors, which of the following must be true about Q?

I. Q is the square of a prime number.
II. Q is the fourth power of a prime number.
III. Q is the product of two prime numbers.

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

For more on factors and factors of perfect squares, check:

http://www.veritasprep.com/blog/2010/12 ... ly-number/
http://www.veritasprep.com/blog/2010/12 ... t-squares/
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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18 Mar 2013, 04:27
Thank you Karishma. !! I do visit Veritasprep and the links you mentioned on this topic was really helpful. Many thanks!!
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Posts: 744
Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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03 Apr 2017, 06:13
emmak wrote:
If Q, a positive integer, has 5 factors, which of the following must be true about Q?

I. Q is the square of a prime number.
II. Q is the fourth power of a prime number.
III. Q is the product of two prime numbers.

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Given: Q>0
Q has total 5 factors. (Odd number of factors)

(1) Not necessary. For example, 5^2=25 but the total no. of factors are 3 (2+1).
(2) YES. For example, 2^4, total no. of factors would be 5 (4+1).
(3) Not possible.
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]

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25 May 2018, 04:02
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Re: If Q, a positive integer, has 5 factors, which of the foll   [#permalink] 25 May 2018, 04:02
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