A Hard Math Problem

[OmgWtfItsALongNameOfDoom]

Ok, this problem was on the AIME (American invitational mathematics examination). I got in two months ago and god, it was hard... Here's a problem from it:

Let N be the number of consecutive 0's at the right end of the decimal representation of the product 1! (factorial) 2!3!4!....99!100!. Find the remainder when N is divided by 1000.

I got this after quite a long time of thinking...it's a 3-digit integer. I'm in Math analysis and you should be able to solve it with Math Analysis knowledge. Good luck.