  We prove that certain basic hypergeometric series truncated at $k=n-1$ have
the factor $\Phi_n(q)^2$, where $\Phi_n(q)$ is the $n$-th cyclotomic
polynomial. This confirms two recent conjectures of the author and Zudilin. We
also put forward some conjectures on $q$-congruences modulo $\Phi_n(q)^2$.
