Mean Divisibility of Multinomial coefficients

Let m_1,…,m_s be positive integers. Consider the sequence defined by multinomial coefficients: a_n=\binom{(m_1+m_2+… +m_s)n}{m_1 n, m_2 n,…, m_s n}. Fix a positive integer k\ge 2. We show that there exists a positive integer C(k) such that \frac{\prod_{n=1}^t a_{kn}}{\prod_{n=1}^t a_n} \in \frac 1{C(k)} \Z for all positive integer t, if and only if GCD(m_1,…,m_s)=1.