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**Probelm**Let p and q be distinct primes. Suppose that H is a porper subset of the integers and H 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, p+q, pq

d. p, p^q, q^p

e. p, pq, p^q

## Homework Equations

The answer is e, as it's given in the book.

## The Attempt at a Solution

My basic strategy here was to try and show it's only closed under addition for e. It's easy to see that adding p to itself q times yields pq, however the same cannot be said for p^q. This didn't seem satisfactory then. My next idea, the one that MIGHT be right, but i'm very much not sure, is that e contains the only terms that are integer factors of p. You know p, pq, and p^q are integers when divided by p, and furthermore they are the only ones that are (since p and q are prime). However, I don't know if this is even signficant, and I certainly don't see the signifigance under a group who's operation is addition.

EDIT - Actually I suppose it's significant in that being an integer multiple it can be seen as "P+P+P+....." an integer number of times. Is that all there is to it? It seems a bit of an odd problem if so.

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