TITLE: Additive group contains exactly three elements of the set $\{p,p+q,pq,p^q,q^p\}$
QUESTION [5 upvotes]: 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 $J$?
I know the solution is $\{p,pq,p^q\}$. However I do not understand why. I thought $J$ can not even be a finite group. Maybe there is serious misunderstanding of the problem. Can anyone give me a hand? Thanks.

REPLY [2 votes]: Hint: if you let $x,y\in J$ with $(x,y)=1$, then $J=\mathbb Z$. You can use the Euclidean algorithm to find $m,n\in\mathbb Z$ so that $mx+ny=1$, hence $\langle 1 \rangle\subset J\implies J=\mathbb Z$.