= Solution
At <periapsis>, $\lambda=\varpi$. If the planet has longitude $\lambda_0$ there, then
$$
\phi_1=p(\varpi-\lambda_0),
$$
so $\phi_1/p$ is the angular displacement of periapsis from the planet, modulo $2\pi/p$. At an <astronomical conjunction>, $\lambda=\lambda_1=\lambda_c$, and
$$
\phi_1=q(\lambda_c-\varpi),
$$
so $\phi_1/q$ is the conjunction longitude measured from periapsis, with the $q$ possible branches differing by $2\pi/q$.
The planet's mean motion is $n_1=(p+q)n/p$. Hence the <synodic period>, or mean interval between conjunctions, is
$$
\boxed{T_{\rm syn}=\frac{2\pi}{n_1-n}=\frac pqT},
$$
where $T=2\pi/n$ is the planetesimal's <orbital period>.
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