Factors of some truncated basic hypergeometric series

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor ${\mathrm{\Phi }}_n(q{)}^2$, where ${\mathrm{\Phi }}_n(q)$ 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 ${\mathrm{\Phi }}_n(q{)}^2$.