Prime Numbers: Proving that a "minimal residue" exists

Question

In a class of mine the lecturer introduced the term of "minimal residue" as follows:

Definition: If $p$ is an odd prime there is just one residue of $n \text{ mod } p$ between $-\frac{1}{2}p$ and $\frac{1}{2}p$. It is called minimal residue.

I do not see why such a minimal residue should necessarily exist or why it should be unique. I tried to google the term but I could not find anything about it. Could you tell me why this definition makes sense?

Answer 1

If $p$ is an odd prime, and $a\in\Bbb Z$ then $a$ is congruent, modulo $p$, to exactly one element of the set $\{-\frac12(p-1),-\frac12(p-3),\ldots,-1,0,1,\ldots,\frac12(p-3),\frac12(p-1)\}$. To see this, consider $a-pk$ where $k$ is the nearest integer to $\frac ap$.

Answer 2

I think you can best understand this with an example. On the number line $$ \ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \ldots $$ you can find a complete residue system modulo $5$ in the usual interval $[0,4]$ and equally well in the interval $[-2, 2]$. The latter has the residues with the smallest absolute values.

Sometimes using that residue system is more convenient. In particular, it can speed up the Euclidean algorithm.