Fundamental period of $x[kn]$
You are correct. The following is what you have shown.
Let $x[n]$ be a periodic sequence with period N :
$$ x[n] ~=~ x[n + rN] ~~~~~,~~~\forall r,N \in Z^+ $$
Define $y[n] = x[M n]$, where $M$ is a positive integer; then
$$ y[n] = x[Mn] = x[Mn + rN] = x[M(n + \frac{rN}{M})] = y[n+K] $$
$y[n]$ will be periodic with $K$, if $\frac{r ~N}{M}$ is an integer. Hence, set $r$ to any value that makes $\frac{rN}{M}$ an integer.
If $N,M$ are already coprime, then setting $r=M$ makes $K=N$ as the period of $y[n]$. If they are not already coprime, then make them coprime $M_c,N_c$, and then set $r=M_c$ to make $K = N_c$ as the period of $y[n]$.
Make $N,M$ coprime by dividing them by their greatest common divisor $g = \text{gcd}(N,M)$ :
$$ N_c = N / g ~~~,~~~~ M_c = M / g $$
Then, for the general case, the period of $y[n]$ is:
$$ \boxed{ K = \frac{N}{ \text{gcd}(N,M) } } $$