Suppose $q$ is a prime power, $r$ is some positive integer. We can construct the field $\mathbb{F}_q$ where there are $q-1$ non-zero elements. The relation $x\sim y \iff y \text { is a (non-zero) scalar multiple of } x$ in the set of nonzero vectors $\mathbb{F}_q^r-\{0\}$ is an equivalence, and $cx=c'x \iff c=c'$ by cancellation law in $\mathbb{F}_q$. Therefore there are equally sized equivalence classes and there are $\frac{q^r -1}{q-1}$ of them, which is an integer.
Now is there a way in elementary arithmetic showing that $\frac{q^r -1}{q-1}$ is an integer?
