Problem arises due overuse of magic - Part (1)

Number Theory Level 4

Ben plays a game: He writes the numbers from 1 to 2004 on a board, selects \(n\) numbers, then writes their sum \(s \bmod {11} \) on the board, casts a magic chant, so that the previous \(n\) selected numbers vanish from the board and starts over again with the numbers, which are left at the board.

  • selecting \(n\) numbers,
  • writing down the \(s \bmod { 11 } \),
  • making the previous \(n\) numbers vanish.

At the end there were just two numbers left. One was 1000. What was the other one?


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