Division Theorem/Half Remainder Version

Question

How to prove the following Division Theorem/Half Remainder Version?

∀ a , b ∈

Z

, b ≠ 0 : ∃ ! q , r ∈

Z

: a = q b

r , −

    |
    b
    |
  /
  2

≤ r <

    |
    b
    |
  /
  2

Answer 1

Hints.

• For uniqueness, use the fact that the only integral multiple of $b$ in $ [-\vert b\vert /2;\vert b\vert /2)$ is $0$.

  For existence:

• Assume that you proved the existence of $q$ and $r$ if $b>0$. How to deduce the existence when $b<0$ ?

• Assume that $b>0$. Write $a=q'b+r', 0\leq r' \leq \vert b\vert.$ If $b/2\leq r<b$, in which interval $r-b$ belongs to ?