Why does $\{ax \mod n: a, x, n > 0 \land \gcd(a, n) = 1 \}$ is a permutation of $\{0, 1, .., n - 1\}$

Question

Why does $\{ax \mod n: a, x, n > 0 \land \gcd(a, n) = 1 \}$ is a permutation of $\{0, 1, .., n - 1\}$? I've seen some proofs, but I can't intuitively see why is that true. I like to think about euclidean algorithm etc. with two sticks of length $a$ and $b$, but even using this I still can't understand why does set of all multiplies of $a \mod n$ is a permutation of $\{0, 1, ..., n - 1\}$

Answer 1

It is enough to show that $$ x \mapsto a x \pmod{n} $$ is injective.

If $a x \pmod{n} = a y \pmod{n}$, then $n \mid a x - a y = a (x - y)$. Since $\gcd(a, n) = 1$, we have $n \mid x - y$, so $x = y$.