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# 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 $\left\{p, p+q, p q, p^{q}, q^{p}\right\}$. Determine which of the following are the three elements in H.a. $p q, p^{q}, q^{p}$.b. $p+q, p q, p^{q}$.c. $p, p+q, pq$.d. $p, p^{q}, q^{p}$,e. $p, p q, p^{q}$ Solution

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Since $H$ is a subgroup of integers, it is of the form

$H = \{n k \mid n \in \mathbf{Z}\}$

for some $k \in \mathbf{Z}$. Now notice that $p$, $pq$, and $p^q$ are all multiples of $p$, so $\textbf{e.}$ is the correct choice. (Other options fail because $p$ and $q$ are distinct primes.)

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