Question

Show that if aand d are positive integers, then there are integers and rsuch that a = dq + r where -d/2 < r < d/2 .

Short answer

$a=dq+r\text{with}-d/2<r\le d/2$

Step-by-step solution

Step 1

First take q to be a div d

Then we know that $0\le r\le d/2$.

If $0\le r\le d/2$, then we have shown that a = dq + r with $-d/2<r\le d/2$.

Otherwise we have that $-d/2<r<d$.

Then $a=dq+r=d(q+1)+r_1$ .

Here $r_1+d=r$.

Hence$-d/2<r_1\le 0$.

Hence we have that in either case $a=dq+r\text{with}-d/2<r\le d/2$.