15. (2004 German Mathematics Competition Question) There is a game where at the beginning, the numbers $1,2, \cdots, 2004$ are written on the blackboard. Each step of the game includes the following steps:
(1) Choose a set of some numbers on the blackboard;
(2) Write the remainder of the sum of these numbers modulo 11 on the blackboard;
(3) Erase the previously chosen numbers.
When the game proceeds to the point where only two numbers remain on the blackboard, one of which is 1000, what is the other number?
15. The other number is 4.

Since the sum of all numbers on the blackboard remains unchanged modulo 11 at each step, and \(1+2+\cdots+2004 \equiv 3(\bmod 11)\), \(1000 \equiv-1(\bmod 11)\), the other number must be congruent to 4 modulo 11. However, during the game, all numbers written on the blackboard are less than 11, and 1000 is one of the original numbers on the blackboard, so the other number must be 4.