Consider the following problem:
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 $J$?
$A. ~pq,~p^q,~q^p$
$B. ~p+q,~pq,~p^q$
$C. ~ p,~p+q,~pq$
$D. ~p,~p^q,~q^p$
$E. ~p,~pq,~p^q$
Here are my questions:
- What properties of ${\bf Z}$ does one need to use here?
- How to solve the problem above?
- Can one generalize this problem with changing the condition "exactly three"?