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

Question

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

Bunuel · Accepted Answer

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.

Bunuel · Answer

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.

mau5 · Answer

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.

tanveer123 · Answer

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 ??

tanveer123 · Answer

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 !!

KarishmaB · Answer

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

tanveer123 · Answer

Thank you Karishma. !! I do visit Veritasprep and the links you mentioned on this topic was really helpful. Many thanks!!

Shiv2016 · Answer

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.

arbazfatmi1994 · Answer

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.

Paras96 · Answer

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

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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

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