Proof on Divisibility of Binomial Coefficients

Question

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$ n \nmid {n \choose i} $$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.

Answer 1

Claim: $\forall \ p$ such that $p$ is a prime, $$m \nmid {m \choose p}$$ where $m = kp$, $k \in \Bbb Z$ and $k \gt 1$

Proof: $${m \choose p} = {m(m-1)\cdots(m-p+1) \over p(p-1)\cdots1}$$

Observe that since $m$ is a multiple of $p$, none of the other $p-1$ integers are divisible by $p \implies $ p should divide $m \implies m \nmid {m \choose p}$.

Also, $m$ cannot be formed again from the remaining integers, since none of them have the factor $p$.