Yes. The product of n odd numbers is an odd number. The product of an odd number and an even number is even. So you only need 1 even number as a factor of a product of integers for the product to be even. That was the clue that shakarri gave you. So you can multiply 2004 odd integers together, getting an odd number. When you multiply that odd number by an even number, you get an even number. For a product of n integers to be even, you can have at most n - 1 odd factors because at least one factor must be even.

Remember this trick of trying simpler problems of the same type when you are stuck. It doesn't always work, but if you are stuck, you need things to try.