Prooving that p devide $C_n^k$ (Related to Fermat's Little Theroem)

Question

Let $p$ be a prime number, and $k$ be an integer with the following property: $1\leq k \leq p-1$. Why can't we just say that $$C^k_n=\frac{p!}{k!\cdot(p-k)!}=p\frac{(p-1)!}{k!\cdot(p-k)!}=pK\implies p/C^k_n$$ for $K\in \mathbb{Z}$?