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Well, lets see if we can go through it from the basics. $x \bmod y$ is the unique integer $x + \ell y$ that satisfies $0 \le x + \ell y < y$, for some integer $\ell$ (which might be positive, negative or zero) I will skip the parts that prove that, if $x, y$ are both integers and $y > 0$, then there exists such an $\ell$ and it is, in fact, unique. ...

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