Why is it that for three numbers $i, j, n \in \mathbb{N}$, if $i \equiv j \pmod{n}$ and $i, j \leq n$, then $i = j$?

Question

For $i, j, n \in \mathbb{N}$, if $i \equiv j \pmod{n}$ and $i, j \leq n$, then $i = j$.

Might be a trivial question, but I don't see why this holds. Can someone explain to me why this is true?

Answer 1

If $i\equiv j\pmod{n}$, then by definition $n\mid i-j$, i.e., $i-j=kn$ for some integer $k$. If, however, $i$ and $j$ are positive integers, and $i,j\le n$, then

$$-(n-1)=1-n\le i-j\le n-1\;,$$

and the only multiple of $n$ between $-(n-1)$ and $n-1$ is $0=0\cdot n$. Thus, $k=0$, and $i=j.$