Divisibility of Product of Consecutive Integers

Theorem
The product of $n$ consecutive positive integers is divisible by the product of the first $n$ consecutive positive integers.

That is:
 * $\displaystyle \forall m, n \in \Z_{>0}: \exists r \in \Z: \prod_{k \mathop = 1}^n \paren {m + k} = r \prod_{k \mathop = 1}^n k$

Proof
Hence the result, and note that for a bonus we have identified exactly what the divisor is:
 * $\dbinom {m + n} m$