Multivariate Rational Functions

Question

Let $P(x_1,\cdots,x_n)$, $Q(x_1,\cdots,x_n)$ be multivariate polynomials with integer coefficients. Assume that for all $(x_1,\cdots,x_n) \in \mathbb{Z}^n$ such that $Q(x_1,\cdots,x_n) \neq 0$ we have: $$\frac{P(x_1,\cdots,x_n)}{Q(x_1,\cdots,x_n)} \in \mathbb{Z}$$Does it necessarily follow that there exists a multivariate polynomial $R(x_1,\cdots,x_n)$ with rational coefficents such that $P/Q=R$