# Division Algorithm problem

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 $$r\geq2b$$.

• 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. – Bill Dubuque May 4 at 9:35
• Oops, above should be $\ldots < b\ \$ – Bill Dubuque May 4 at 10:08