On the polynomial $frac{1}{[n]_q}left[{n atop k} right]_q$

In the work The Friedman–Joichi–Stanton Monotonicity Conjecture at Primes, George Andrews gave a proof of the fact (first mentioned by Jim Propp a while ago) that the q-binomial coefficient View the MathML source divided by the q-integer [n]q is a polynomial in q, as long as n and k are relatively prime (see [G.E. Andrews, The Friedman–Joichi–Stanton Monotonicity Conjecture at Primes, in: DIMACS Ser., Amer. Math. Soc., in press, Theorem 2]). In this note we provide a proof that permits to generalize Theorem 2 in the case in which n and k are not relatively prime, and further, to extend Theorem 2 to the q-multinomial coefficient.