= Solution
{c}
A return path with $u$ upward steps and $v$ downward steps satisfies $au=bv$. Since $a,b$ are <coprime>, this is equivalent to $u=bk$ and $v=ak$ for an <integer> $k\geq1$. Its length is therefore $k(a+b)$.
Conversely, for each such $k$, any ordering of $bk$ upward and $ak$ downward steps is an allowed positive-<probability> return path. The possible strictly positive return times are exactly the positive multiples of $a+b$. By the definition of the <period of a state in a Markov chain> as their <greatest common divisor>,
$$
\boxed{\text{Every state has period }a+b\geq2.}
$$
In particular, none of the states is <aperiodic>.
Back to article page
