After reading about periodicity of states in Markov chains I have problems understanding intuitively the aperiodic states.
Following the Wikipedia definition if the return to one state can be done in $3n$ and $5n$ steps ($n\in \mathbb{N}$), then the period will be g.c.d{3,5} = 1. So the chain will be aperiodic. I found it counterintuitive as the return can only be done in $3n$ or $5n$ steps. That seems to me a periodic (informally speaking) behavior.
This source uses the Wikipedia definition of period but a more intuitive (to me) definition of aperiodic state. My problem here is that I cannot fit my example into that definition.
Can someone help me to understand this? Please, consider that I'm not a mathematician