RMO 2001 ; Part - 2

We can write the difference between the first 1000 positive even numbers and the first 1000 positive odd numbers as: $2\times4\times...\times2000 - (2001-2000)(2001-1998)...(2001-2)$ All of the terms from the expansion of the brackets are divisible by 2001 except for $2000\times1998\times...\times2$ , which is just the product of the first 1000 even numbers. Therefore this term is cancelled out, meaning every term is now divisble by 2001, so the whole thing is divisible by $\fbox{2001}$

We can find the divisibility of this difference by $mod$ function.

$(2 \times 4 \times 6 \times ........\times 1998 \times 2000) - (1 \times 3 \times 5 \times ........\times 1997 \times 1999) mod$ $1997 = 761$ (Not divisible)

$(2 \times 4 \times 6 \times ........\times 1998 \times 2000) - (1 \times 3 \times 5 \times ........\times 1997 \times 1999) mod$ $1999 = 1$ (Not divisible)

$(2 \times 4 \times 6 \times ........\times 1998 \times 2000) - (1 \times 3 \times 5 \times ........\times 1997 \times 1999) mod$ $2003 = 1000$ (Not divisible)

$(2 \times 4 \times 6 \times ........\times 1998 \times 2000) - (1 \times 3 \times 5 \times ........\times 1997 \times 1999) mod$ $2001 = 0$ (Divisible)

Thus, the answer is $\boxed{2001}$