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

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

Please check my signature for the link to the relevant blog posts.

Originally posted by KarishmaB on 18 Mar 2013, 02:34.
Last edited by KarishmaB on 11 Oct 2022, 02:24, edited 1 time in total.
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]
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]
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]
If a number has 5 factors, that gives us a hint that it only has one distinct prime factor because the number of factors is (p+1)(q+1)(r+1)

But 5 is a prime number...

so it only has one term, i.e (p+1)

and P+1 = 5, P=4

So, it has to be a fourth power of a prime number.
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Re: If Q, a positive integer, has 5 factors, which of the foll [#permalink]
Given, Q, a positive integer, has 5 factors

Lets take the following cases one by one

I. Q is the square of a prime number. Q = P^2 => Q has (2+1)=3 factors therefore wrong
II. Q is the fourth power of a prime number. Q = P^4 => Q has (4+1)=5 factors therefore correct
III. Q is the product of two prime numbers. Q = P*R => Q has (1+1)(1+1)=4 factors therefore wrong

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