Can this be done using Sylow theorems?

Question

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers and $H $is a group under addition that contains exactly three elements of the set {$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. $p,p+q,q^p$

c. $p,p^q,q^p$

d. $p,pq,p^q$

Answer 1

Hint: the subgroups of $\mathbb{Z}$ have the form $n\mathbb{Z}$ for some non-negative integer $n$.

Answer 2

Hint: If $H$ contains two relatively prime integers, then it contains all the integers.