1. ## Division Algorithm Problems

Hello, I need help with the following problem.

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

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

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

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