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For positive integers $k,n$ , is it true that $k$ divides $\binom{kn}{n}$?

I can write $$\binom{kn}{n}=\frac{(kn)(kn-1)\cdots(kn-n+1)}{n(n-1)\cdots 1}$$ but must the $k$ at the top remain after cancelling stuff out?

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Note that $$\binom{nk}n=\frac{nk}n\binom{nk-1}{n-1}=k\binom{nk-1}{n-1}\;,$$

and $\binom{nk-1}{n-1}$ is an integer.

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