Let $n,\alpha\in\mathbb{N},\beta\in\mathbb{N}_0$, and let $p$ be odd prime number s.t. $p^\beta|n$.
How do we prove that $p^{\alpha+\beta+1}|{n\choose k}p^{k\alpha}$ for every $k\in\{1,2,\ldots,n-1\}$?
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