jmc

Let $n=2m$. Let $b$ be the number of cars that Ray originally had. Clearly $b=6a$ for some positive integer $a$. Additionally, $$\begin{array}{rl}b-2\equiv 0(\mathrm{mod}n)& ⟹6a\equiv 2(\mathrm{mod}2m)\\ & ⟹3a\equiv 1(\mathrm{mod}m).\end{array}$$ Such $a$ exists if and only if $3$ is invertible modulo $m$. In other words, $\gcd (3,m)=1$. We have that $n<10⟹m=\frac{n}{2}<5$. The only $0<m<5$ relatively prime to $3$ are $1,2,4$. Thus, there are $\boxed{3}$ possible values of $n$: $2$, $4$, and $8$.