I'd like you to check my work. I have this nagging feeling that I am missing something, or worst yet, that my proof has some error I can't see.
(b) For which pairs is by well defined?
We claim that this is true for pairs iff . Thus we need to show .
( ): Assume . Then for some . So, for some .
( ): For the converse, we use the contrapositive. Assume . Then by the Division Algorithm, , , for unique . So (which is an integer by hypothesis). Thus, , for some , which implies for any . Thus, , which means .
I don't like the second part of the proof too much, the part where i said not equal to nl mostly. Plus, this is just an observation of mine that I saw and proved, how do I know these are the only m and n for which this works?