If p and q are positive integers such that pq has exactly 4 unique pos

If p and q are positive integers such that pq has exactly 4 unique positive factors and \(p < q\) , what is the value of the integer p?

Given that pq has exactly 4. We can have the following two cases:

(i) \(p = 1\) and \(q = (prime)^3\) --> \(pq = 1*(prime)^3\) --> the number of factors = 3 + 1 = 4. For example, \(pq = 1*2^3 = 8\).

(ii) \(p = prime_1\) and \(q = prime_2\) --> \(pq = prime_1*prime_2\) --> the number of factors = (1 + 1)(1 + 1) = 4. For example, \(pq = 2*3 = 6\).

As we can see p is either 1 or some prime.

(1) The sum of p and q is an odd integer. For the sum of two integers to be odd one of them must be odd and another must be even. We cannot get the unique value of p. For example:
p = 1 and q = 2^3 = 8
p = 2 and q = 3.

Not sufficient.

(2) The difference between p and q is 1. Can we have case (i) here? No. If p = 1 and \(q = (prime)^3\), the the least difference between p and q is 2^3 - 1 = 7. Therefore both p and q must be primes. The only primes with difference of 1 are 2 and 3. Sufficient.

Answer: B.

Hope it's clear.