Question:
Find the smallest $n$ for which there exists a Markov chain with $n$ states such that it is periodic simultaneously with period $4$ and $6$? Put "does not exist" if such a Markov chain does not exist.
The definition of periodicity I use:
State $a$ is periodic if there exists an integer $d>1$ (called the period of $a$) such that if the probability of returning via a path of length $k$ is greater than zero for some $k$, then $d|k$.
A Markov chain having no periodic states is called a non-periodic chain.
My answer is $n = 12$, because we can create a chain as below (cycle that can be traversed in only one direction):
Then, every return path has a length of $12 = lcm(4,6)$.
Since $4$ and $6$ divide $12$ so both of them can be a period.
Is this a correct solution?