Largest Divisor of $n$ that divides $\binom{n}{r}$

Hermite's First Divisibility Property for Binomial Coefficient $\binom{n}{r}$ states, for $n,r \geq 1$ ,

$\frac{n}{gcd(n,r)} | \binom{n}{r}$

Here $gcd(8778,1105)=1$ , so $8778| \binom{8778}{1105}$ . As the largest divisor of $8778$ is $8778$ itself, the answer is $8778$ .

This is my argument. But I believe there must be some better argument, even just by using Hermite's First Divisibility Property . I'm still working on that.

the solution is very simple . the factors of 8778 are 2,3,7,11,19 .after substitution we see that the numerator has 8778 factorial which already contains 8778 . thus the largest divisor of 8778 that divides 8778 factorial is 8778.