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Prove that the expression $$\frac{\gcd(n,m)}{n}\binom{n}{m}$$ is an integer for all pairs of integers $n\ge m \ge1$.

My work so far:

If $\gcd(m,n)=1$ then $$\frac{\gcd(n,m)}{n}\binom{n}{m}=\frac1n\cdot \frac{n!}{m!(n-m)!}=\frac{(n-1)!}{m!(n-m)!} \in \mathbb Z$$ If $\gcd(m,n)=d>1$

I need help here

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Roman83
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    Even in the gcd=1 case, it's not immediately obvious that dividing $\binom{n}{m}$ by $n$ gives an integer. [Just saying that requires a proof.] – coffeemath Nov 11 '16 at 14:48

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