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

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

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New post Updated on: 12 Mar 2013, 01:14
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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 11 Mar 2013, 23:33.
Last edited by Bunuel on 12 Mar 2013, 01:14, edited 1 time in total.
Edited the question.
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Re: If Q, a positive integer, has 5 factors, which of the foll  [#permalink]

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New post 12 Mar 2013, 01:22
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).

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

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New post 12 Mar 2013, 01:42
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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New post 17 Mar 2013, 09:14
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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New post 17 Mar 2013, 09:50
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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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New post 17 Mar 2013, 10:44
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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New post 18 Mar 2013, 01:34
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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New post 18 Mar 2013, 03:27
Thank you Karishma. !! I do visit Veritasprep and the links you mentioned on this topic was really helpful. Many thanks!!
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Re: If Q, a positive integer, has 5 factors, which of the foll  [#permalink]

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New post 03 Apr 2017, 05:13
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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Re: If Q, a positive integer, has 5 factors, which of the foll &nbs [#permalink] 25 May 2018, 03:02
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