Tiling Into Slanted Rectangles

This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years.) It is seen as a possible follow on from

"Tiles in the Garden".

This activity takes "Tiles in the Garden", much further. We can keep the main ideas the same - 

• Square tiles
• A corner of a tile at each corner of the rectangle
• The ability to slice a tile into parts so as to use each part

So this one used $26$ and the slope was generated by going along $1$ and up $5$.

This time let's put on a limit of using less than $100$ tiles.

What sizes of rectangles could be filled obeying the three rules?

How many tiles for each rectangle you find?

Are there any numbers of tiles between $10$ and $100$ for which there cannot be a rectangle?