Question 648576
Let x and y be positive integers.

Then let x*y be maximized by  x+y = 2005

In general, the closer you get to a square, the larger the product will be. We can either use this fact, or we can make a table. I'll leave you to make a table if you want. Knowing that 2005 is odd, x and y can differ by at most one. 

x + x+1 = 2005

2x = 2004

x = 1002

(1002+1) is the other. 

Then 1002 and 1003 are the two integers.

Their product is 1005006.