A hole new board game puzzle

Can you cut a 3×8 rectangle into 2 pieces, then rearrange to form a 2×12 rectangle?

Here's the solution.

Now let's consider a generalization. Suppose you have two rectangles, one with dimensions a×b, the other, c×d. Both rectangles have exactly the same area. Find some examples of a×b rectangles that can be cut into exactly 2 pieces to yield a c×d rectangle. Can you find any patterns in your examples? What about cutting the rectangle into \(n\) pieces? What about cutting and rearranging cuboids instead?