Question 49
The numbers $1, 2, \ldots, 2009$ are written on a board. Some of them are erased and, in their place, the remainder of the sum of the erased numbers upon division by $13$ is written on the board. After a finite number of repetitions of this procedure, only three numbers remain on the board, two of which are $99$ and $999$. Determine the third number that remains on the board.