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Thread: Show that there exists a unique element c of Z such that n divides ab-c

  1. #1
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    Exclamation Show that there exists a unique element c of Z such that n divides ab-c

    Let n be a positive integer. Define Zn = {0, 1, 2, ..., n-1}. Let a, b ∈ Zn. Show that there exists a unique element of c of Zn, such that n divides ab-c.

    OK. This is the question translated in words. Let n be a positive integer. Define Z sub n to be equal to the set {0, 1, 2, ..., n-1}. Let a, b be elements of set Z sub n. Show that there exists a unique element of c of Z sub n, such that n divides (a times b) minus c.

    This question I am beyond lost, especially since our prof just gave us this problem with no examples to compare it to. I feel like you have to take some arbitrary element and somehow manipulate via proof to show that n divides ab-c.
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    Re: Show that there exists a unique element c of Z such that n divides ab-c

    we divide $\displaystyle ab$ by $\displaystyle n$ to get a quotient $\displaystyle q$ and remainder $\displaystyle c$
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    Re: Show that there exists a unique element c of Z such that n divides ab-c

    But how does this relate to the original question of showing that "there exists a unique element"? Or is what you have there sufficient?
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    Re: Show that there exists a unique element c of Z such that n divides ab-c

    $\displaystyle a b=n q+c$

    $\displaystyle 0\leq c < n$

    $\displaystyle c\in \mathbb{Z}_n$ and $\displaystyle n$ divides $\displaystyle a b - c$

    $\displaystyle c$ is the number we are looking for

    we need to show that it is the unique such number
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