= Solution
\b[True.] For a state $i$, let
$$
\mathcal R_P(i)=\{n\geq1:(P^n)_{ii}>0\}
$$
be its possible return times under $P$, and define $\mathcal R_Q(i)$ similarly. Every positive $P$-path is a positive $Q$-path, so
$$
\mathcal R_P(i)\subseteq\mathcal R_Q(i).
$$
The greatest common divisor of the larger set divides that of the smaller set. Since the latter is one by the assumed <aperiodic Markov chain> property, the former is also one. Hence every state is aperiodic under $Q$.
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