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