Greatest Common Divisor Integer

Question

I need help with the following problem:

Let $n$ and $m$ be positive integers.

Prove that

$$\frac{\gcd(n,m)}{n}{n \choose m}$$

is an integer.

Answer 1

Let $d =\gcd(m,n)$. By the Euclidean algorithm, there exist integers $A$ and $B$ such that $d = Am + Bn$. Thus

$$\displaystyle d \binom{m}{n} = A m \binom{m}{n} + Bn \binom{m}{n} = A m \binom{m}{n} + B m \binom{m-1}{n-1}$$, by the absorption identity. Then

$$\displaystyle d \binom{m}{n} = m \left[ A \binom{m}{n} + B \binom{m-1}{n-1} \right] = mC$$, where $C$ is an integer.

Thus $\displaystyle \frac{m}{d}$ divides $\displaystyle \binom{m}{n}$. In other words, $\displaystyle \frac{m}{(m,n)}$ divides $\displaystyle \binom{m}{n}$