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### Problem 10

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers that 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) $p+q,pq,p^{q}$

(c) $p,pq,p^{q}$

(d) $p,p+q,pq$

(e) $p,p^{q},q^{p}$

**Solution**

Use relative primality: if any two elements are relatively prime, we may write a Z-linear combination of this pair as $1$. Then, we can easily generate the elements not picked in the list.

For us, we can do this with all choices but part (c). (All elements in part (c) are multiples of $p$; hence you can't generate $q$ because $gcd(p,q) = 1$.)

So, only (c) can form a group $H$.

As for the other parts:

(a) Since $gcd(p,q) = 1$, $gcd(p^{q},q^{p}) = 1$. Thus, $1$ is in $H$. Since $1$ is in $H$, any multiple of $1$, such as the elements not used from your given set are also in $H$.

(b) Since $gcd(p,q) = 1$, $gcd(p+q, p) = 1$ and thus $gcd(p+q, p^{q}) = 1$. Thus, $1$ is in $H$.

(d) Since $gcd(p,q) = 1$, $gcd(p+q, p) = 1$. Thus, $1$ is in $H$.

(e) See part (a).