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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"?
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    Sorry, @Gerry: it was directed to OP. I understood you per$f$ectly well!2011-06-30

2 Answers 2

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You need to use the fact that if $a,b$ are relatively prime then they generate all integers.

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    Hmm, Bezout's identity.2011-07-06
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You don't need any properties of $\mathbb{Z}_n$. What you need is the fundamental theorem of arithmetic (unique prime factorization of integers) and the following property of the additive group $\mathbb{Z}$: the subgroups of $\mathbb{Z}$ are precisely the subsets $n\mathbb{Z}$ $(n\in\mathbb{Z})$. To prove this fact, show that if $a$ and $b$ are in your subgroup, then $\gcd(a,b)$ is in your subgroup.