1. Division Algorithm Problems

Hello, I need help with the following problem.

Use the division algorithm to prove that if $\displaystyle a$ and $\displaystyle b$ are integers with $\displaystyle b \neq 0$, then there exist unique integers $\displaystyle q$ and $\displaystyle r$ such that $\displaystyle a=bq+r$, with $\displaystyle 0 \leq r < |b|$

2. Originally Posted by akolman
Hello, I need help with the following problem.

Use the division algorithm to prove that if $\displaystyle a$ and $\displaystyle b$ are integers with $\displaystyle b \neq 0$, then there exist unique integers $\displaystyle q$ and $\displaystyle r$ such that $\displaystyle a=bq+r$, with $\displaystyle 0 \leq r < |b|$

Say $\displaystyle a = bq_1+r_1 = bq_2 + r_2$.
Then, $\displaystyle b(q_1-q_2) = (r_2 - r_1)$. [1]
However, $\displaystyle 0 \leq r_2 < |b| \implies -r_1 \leq r_2 - r_1 < |b| - r_2 \implies - |b| < r_1 \leq r_2 - r_1 < |b| - r_2 < |b|$.
This means $\displaystyle |r_2-r_1| < |b|$.
Look at [1] and take absolute values, $\displaystyle |b||q_1 - q_2| = |r_2 - r_1|$.
Now if $\displaystyle q_1\not = q_2 \implies |q_1-q_2| \geq 1$ and so $\displaystyle |b||q_1-q_2| > |r_2-r_1|$ which is a contradiction.
This forces $\displaystyle q_1=q_2$ which ultimately forces $\displaystyle r_2 = r_1$.