KöMaL 2021 C1660

The positive integers $1$ to $61^2$ are written in the fields of a $61\times61$ chessboard, starting from the top left corner and proceeding along each row in succession. Then some changes are made as follows. In the first move, the sign of each number is changed to negative. In the second move, the signs of all even numbers are changed. In the third move, the sign of every multiple of 3 is changed, and so on, while the moves are meaningful. When all this is completed, how many $1\times2$ rectangles will there be on the chessboard in which the sum of the numbers is negative?