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