Prime Factor

If $p$ is prime and $1000 > p > 661$ , then $3p > 2000$ and so the only multiples of $p$ between $1$ and $2000$ are $p$ and $2p$ . Thus the index of $p$ in $2000!$ is $2$ . In addition the index of $p$ in $1000!$ is $1$ , and hence the index of $p$ in $\binom{2000}{1000}$ is $0$ , meaning that $p$ is not a factor of $\binom{2000}{1000}$ . On the other hand $3 \times 661 = 1983< 2000$ , and so the index of $661$ in $2000!$ is $3$ , while the index of $661$ in $1000!$ is $1$ . Thus the index of $661$ in $\binom{2000}{1000}$ is $1$ , and hence $\boxed{661}$ divides $\binom{2000}{1000}$ , and so is the largest $3$ -digit prime factor of $\binom{2000}{1000}$ .

Mathematica

Max@Select[First /@ FactorInteger@Binomial[2000,1000],#<1000&]

returns 661