Prime factor for selection!

$\dbinom {2000}{1000} = \dfrac {2000!}{1000! \times 1000!} = \dfrac {2000\times 1999 \times 1998 \times \cdots \times 1001}{1\times 2 \times 3 \times \cdots \times 1000} = \dfrac {\prod_{n=1001}^{2000} n}{1000!}$

We note that since $\binom {2000}{1000}$ is an integer. the numerator $\prod_{n=1001}^{2000}n$ is divisible by the denominator $1000!$ . Every 3-digit prime in the numerator is cancel once with the one in the denominator. Any remaining 3-digit prime $p$ must be $3p < 2000$ or $p < 666$ . The largest of which is $\boxed{661}$ .