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

1. ## 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.

2. ## 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 $\displaystyle c$

3. ## 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?

4. ## 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