How many elements of $(\mathbb{Z}/p\mathbb{Z})^n$ are scalar multiples of a fixed non-zero $n$-tuple $x$? Imagine that you have a fixed, non-zero $n$-tuple $x$, such that values in $x$ belong to $(\mathbb{Z}/p\mathbb{Z})^n$, where $p$ is prime. In other words, if, for example, our prime $p$ was $7$ and the value of $n$ was $3$, a possible value of $x$ could be
$$
(1, 5, 3).
$$
The actual question is the following:
How many elements on the aforementioned set $(\mathbb{Z}/p\mathbb{Z})^n$ are scalar multiples of $x$?
This is not one of my homework problems, although I am currently getting into Abstract Algebra through one of my courses. I would appreciate any and all help, since I have been trying to figure out a nice way to approach this for quite some time now.

The answer is $p$ because if $av = bv$ we have $(a-b)v=0$. If you look at one of the coordinates where $a$ is not $0$ you will see that $a-b$ must be $0$ and so we are done.
