When $HCl$  gas is passed through a saturated solution of common salt, pure $NaCl$  is precipitated because

The solubility product of $NaCl$  is lowered by $HCl$

$HCl$ is highly ionized in solution

$HCl$  is highly soluble in water

The ionic product of $\left[N{a}^{+}\right]\left[C{l}^{-}\right]$  exceeds the solubility product of $NaCl$