Can anyone give me some hints on the following problem:

For integers x and y, show that if and only if and

Thanks in advance

Shinn

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- April 23rd 2009, 02:26 AMshinnDivisiblity Proofs
Can anyone give me some hints on the following problem:

For integers x and y, show that if and only if and

Thanks in advance

Shinn - April 23rd 2009, 03:22 AMSoroban
Hello, Shinn!

Quote:

For integers and , show that: .

Since

Since

Then: .

Hence: .

Therefore: .

- April 23rd 2009, 03:30 AMshinn
Hi, thanks for the reply.

Is it correct if i prove "If then and " by the following:

Given

Then, and

(how would I write out a reason for this line?)

Thus, and - April 23rd 2009, 10:01 AMclic-clac
Hi

I don't see any very easy proof.

Maybe you can do that using a contraposition proof:

assume doesn't divide nor i.e. there are integers such that and

Then where

therefore i.e. does not divide

To prove compute all squares in

Another way to show that would be to say that is a factorial ring, that is an odd prime and so it is irreducible in and as a consequence