Let p and q be distinct primes. There is a proper subgroup J of the additive group of integers which contains exactly three elements of the set {p,p+q,pq, p^q, q^p}. Which three elements are in J?

(A) pq,p^q,q^p

(B) p+q,pq,p^q

(C) p,p+q, pq

(D) p, p^q,q^p

(E) p,pq,p^q

Ans:(E)

The only relevant idea I can think of is using Fermat's little Theorem:

Since p and q are distinct primes,then p^q = p(mod q)

which means p^q-p=nq where n could be any integers and of course pq is one of the

element.

But since it's a subgroup of additive group of integers, where is the identity and inverse?

Thanks. Any idea for this question?