If a and $b> 0$ are integers, then prove that there exists unique $q$ and $r$ such that $a= qb + r$ where $2b \leq r <3b$.

I have attempted this problem by putting $q= q-2$ and proceeding similar to the proof of division algorithm. But I cannot reach the conclusion $r\geq2b$.

  • $\begingroup$ HInt: Shift $\, 0 \le a-qb \le b\,$ by $\,2b.\ \ $ This is a dupe and will likely soon be closed as such. Better to delete it if you see the answer from the hint. $\endgroup$ – Bill Dubuque May 4 at 9:35
  • $\begingroup$ Oops, above should be $\ldots < b\ \ $ $\endgroup$ – Bill Dubuque May 4 at 10:08

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