Lattice Point Distribution by a Diagonal Line in a Rectangle

Let $a, b$ be positive integers. In the Cartesian coordinate plane, consider the rectangular region $S$ (including the boundary) enclosed by the points $(1,1)$, $(1,b)$, $(a,1)$, and $(a,b)$. The line $$ l: (2a+1)y - (2b+1)x = 0 $$ divides $S$. Let $f(a,b)$ be the absolute value of the difference between the numbers of lattice points on the two sides of the line $l$ inside the rectangle $S$. How to prove that:

(1) Let $|a-b| = k$. $$ f(a,b) = f(a + nk, b + nk), $$ where $n$ is a non-negative integer;

(2) For any positive integers $a, b, c, d$ satisfying $$ |a-b| = |c-d| = a + c + 1, $$ then $$ f(a,b) + f(c,d) = a + c. $$

(1) and (2) correspond to periodicity and symmetry, respectively. Also, I would like to gain a deeper and more essential understanding of this problem.