There are a few questions I'm having about some proofs in number theory:
- I'm trying to prove the division theorem, that there exist unique integers q and r such that , where . The proof was broken up into cases where a > 0, a = 0 and a < 0, all with b > 0. I have to complete the proof with b < 0. I know this involves saying that -b > 0 and just replacing b with -b in the other cases, but do I have to do three more cases or just one?
- I'm trying to prove that the r and q are unique. Say there are two of each, r and r', and q and q'. Then a = bq + r = bq' + r', so b(q - q')=r - r'. I have in my lecture notes that the LHS is a multiplier of b (which is obvious) but also that -b < r - r' < b (I don't understand how this was established).
For the first one, you should be able to get away with 1 case. (Replace a with -a and b with -b and then use the result already proved).
For your second question. Since r < b, and r' >=0, r -r' <b. Similarly for the other inequality.
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