Finalized # 23

23. (Preliminary for T, March 27; finalized Sat, March 31) Prove uniqueness of the quotient and remainder in the division algorithm.
(Write clearly and don't make jumps that are too large. If you find yourself struggling to mentally justify a claim you make in the proof, prove a little lemma and then use it.)

Pf:
Suppose $0\leq r_1 \leq |b|$ and $0\leq r_2 \leq |b|$.
Then:
$a=q_1 b+r_1$ and $a=q_2 b+r_2$
By setting both a to be equal to eachother, then
$q_1b+r_1=q_2b+r_2$
$\Longrightarrow b(q_1-1_2)=r_2-r_1$
If $r_2-r_1=0, \longrightarrow r_1=r_2$
Since $b\neq 0$, then if $r_1=r_2, \longrightarrow q_1-q_2 =0$.
Then $q_1=q_2$.
Thus the quotient and the remainder are unique.