$a\cdot x \mod n = 0$, if I know $a$ and $n$, what can I say about $x$?

Question

I have the equation

$ax \mod n = 0$

$a,x,n$ are integers and only $x$ is unknown.

Are there any properties of $x$ that follow from this equation?

I tried the following myself, but I am stuck:

Because of the $\mod n $ it follows that $ax = pkn$ with $k,p$ being other unknown integers, thus:

$x=p\frac{kn}{a}$

And this means that I know that $x$ has the property that it can be divided by $\frac{kn}{a}$.

But is that all that I can derive? And how can I calculate $\frac{kn}{a}$?

Answer 1

$x = k\dfrac{n}{a}$, where $k \in \mathbb{Z}$

you said you know $n$ and $a$... just put $k$ as any integer to get a value for $x$.

and since you wanna restrict $x$ to be an integer you must find $k \in \mathbb{Z}: k\dfrac{n}{a} \in \mathbb{Z}$

also if you are looking for the minimum value of $x: x>0$ then set $k = \dfrac{a}{\gcd(n,a)}$