Which three elements are in the proper subgroup H?

[Benzoate]

Homework Statement

Let p and q be distinct primes. Suppose that H is proper subset of the integers nd H is grou under addition that contains exactly three elements of the set {p,p+q,pq, p^q, q^p}. Determine which of the following are the three elements in H:
a) pq, p^q, q^p
b)p+q, pq,q^p
c) p, p+q,pq
d) p, p^q,q^p
e)p,pq, p^q

Homework Equations

no equations for this problem

The Attempt at a Solution

The back of my textbook says the answer is e, but I thought it would be c . I don't understand why the answer is e because if a group is under addition, the additive properties of the group should be p+q . the properties for a group under multiplication would be p*q.
a:

 

[matt grime]

p*q ia p added to itself q times. So obviously if p is in a subgroup of the integers with addition, then so is p*q.

It can't be c. If it were c, then it would contain co-prime integers, hence all integers, but you are told it is a *proper* subgroup.