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Math Help - 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 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|

    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 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|

    Thanks in advance
    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.
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