How to prove a product of k consecutive integers is always a multiple of k?

Question

How to explain and prove that a product of k consecutive integers is always a multiple of k!.

Answer 1

Hint: $(n+1)(n+2) \dots (n+k) = k!$ $\left( \begin{array}{c} n+k \\ n \end{array}\right)$

Answer 2

Hint:

For each $n$ there are $k-1$ numbers $m$ with: $nk<m<(n+1)k$.