If you were told to draw a rectangle along the lines of a sheet of graph paper such that its area is 40 squares, you could choose rectangles measuring 8x5, 10x4, 20x2 or 40x1.

For two of these, 8x5 or 10x4, you would find that you could draw a diagonal across the rectangle that would pass through exactly 12 squares.

What is the smallest number of squares that could be the area of three different rectangles whose diagonals pass through the same number of squares? How many squares does this diagonal pass through?