A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n$.
To solve this problem, we need to determine the maximum number of cells \( n \) that can be marked on a \( 110 \times 110 \) grid such that the distance between any two marked cells is not equal to 15 moves of a chess king. 

1. **Understanding the Distance Metric**:
   The distance between two cells \( A \) and \( B \) on a chessboard is defined as the minimum number of moves a king needs to move from \( A \) to \( B \). The king can move one square in any direction (horizontally, vertically, or diagonally). Therefore, the distance between two cells \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   \max(|x_1 - x_2|, |y_1 - y_2|)
   \]

2. **Coloring Argument**:
   We can use a coloring argument to simplify the problem. Consider coloring the \( 110 \times 110 \) grid in a checkerboard pattern. This means that no two adjacent cells share the same color. Since a king's move changes the parity of the cell (from black to white or white to black), any two cells that are 15 moves apart must have the same color.

3. **Maximal Number of Marked Cells**:
   By marking all cells of one color (say black), we ensure that no two marked cells are 15 moves apart because a king's move always changes the color. The number of black cells in a \( 110 \times 110 \) grid is:
   \[
   \left\lceil \frac{110 \times 110}{2} \right\rceil = \left\lceil 6050 \right\rceil = 6050
   \]

4. **Verification of Maximality**:
   To show that this is the maximal number, we need to prove that marking more than 6050 cells would force at least two cells to be 15 moves apart. Consider the following:
   - If we mark more than 6050 cells, we would have to mark at least one cell of the opposite color (white).
   - Since the grid is \( 110 \times 110 \), marking more than 6050 cells would mean marking at least one white cell.
   - Any two cells that are 15 moves apart must have the same color, and since we have more than 6050 cells, at least one pair of cells would be 15 moves apart, violating the condition.

5. **Conclusion**:
   Therefore, the greatest value of \( n \) such that the distance between any two marked cells is not equal to 15 is 6050.

The final answer is \(\boxed{6050}\).