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.  Choose the correct choice from the following options
A) I only  B) III only  C) II 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=k4∗p4(kp)4=k4∗p4. Now the number of factors of Q = (4+1)*(4+1)= 
<gadget id="calculator">(4+1)*(4+1)</gadget>
<output>25</output>
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.   The Answer is, III only.
<result>B</result>
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.