# Let P be the transition matrix of a Markov chain, and there exists an integer $r \geq 0$ such that every entry of $P^r$ is positive

I would like to show this assumption implies that the Markov chain is irreducible and aperiodic. I was able to show the irreducible part but I'm having trouble proving the aperiodicity. I tried to prove there must exist an odd cycle, but I couldn't seem to proceed. Any help?

Aperiodicity is immediate from the fact if a set $$A$$ of integers contains two consecutive integers then its gcd is $$1$$. Note that $$P^{r+1}$$ also has all entries positive as shown below:
Suppose $$M$$ is a stochastic matrix . Suppose some element $$\sum_k m_{ik} P^{r}_{kj}$$ of the product $$MP^{r}$$ is $$0$$. Then $$m_{ik}=0$$ for all $$k$$ contradicting the fact that $$\sum_k m_{ik}=1$$. Apply this with $$M=P$$.