I need some help with this problem, can you help me?
Given $k\in\mathbb{Z}^*$ prove $\forall x\in\mathbb{Z}, k\mid x(x+1)(x+2)\cdots(x+k-1)$.
Sorry that one is pretty easy, I made a mistake.
The problem is:
[Update] Given $k\in\mathbb{Z}^*$ prove $\forall x\in\mathbb{Z}, k!\mid x(x+1)(x+2)\cdots(x+k-1)$.
HINT: $x,(x+1),(x+2),\cdots(x+k-1)$ are $k$ consecutive integers.
If $x$ is a multiple of $k$ then you are done. But if not so then by division algorithm there is $q,r \in \Bbb Z$ such that $x = kq +r$ and $0 \leq r <k$.
Now consider $k-r$ and see that $x+k-r$ is among the list and is divisible by $k$.
Consider the Binomial coefficient $$\binom{x+k-1}{x-1}$$ and see your question follows. As It is always an integer and $$\binom{x+k-1}{x-1} = \frac{(x+k-1)!}{(x-1)! \times k! } = \frac{x(x+1)(x+2)\cdots(x+k-1)}{k!}$$
Thus $\forall x\in\mathbb{Z},\ \ k!\mid x(x+1)(x+2)\cdots(x+k-1)$.