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6 August, 01:03

Assume a and b are both integers and a > 0. Define a remainder after the division of b by a to be a value r such that r ⥠0, r < a, and there exists an integer q for which b = aq + r. a) Prove uniqueness. That is, if r1 and r2 are both remainders after the division of b by a, then r1 = r2. Y

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  1. 6 August, 01:05
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    r1=r2

    Step-by-step explanation:

    Initially, let's assume there are two remainders r1, and r2. This must fit the following equations:

    b=a. q + r1 (equation 1)

    b=a. q + r2 (equation 2)

    Because we have the same variable b in both equations, we can make equal both expressions, and solving:

    a. q + r1 = a. q + r2

    - r2 = - r2

    a. q + r1 - r2 = a. q + 0

    a. q + r1 - r2 = a. q

    -a. q = - a. q

    0 + r1 - r2 = 0

    r1 - r2 = 0

    +r2 = r2

    r1 + 0 = r2

    r1 = r2 And this is what we want to proof.
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