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Thread: Division Algorithm Problems

  1. #1
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    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 $\displaystyle 0 \leq r < |b|$

    Thanks in advance
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  2. #2
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    Quote Originally Posted by akolman View Post
    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 $\displaystyle 0 \leq r < |b|$

    Thanks in advance
    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 $\displaystyle r_2 = r_1$.
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