Bob and Sarah decide to pave their bathroom floor with ceramic tiles. The bathroom measures $1.4$m by $3.7$m. A tile is a square slab with side length $30$cm. Tiles not closest to the walls of the bathroom must be whole (i.e. not cut into). Only tiles closest to the walls can be rectangular slabs that were made from a $30\times30$ tile, that was cut along a straight edge (by the manufacturer) no more than once. What is the least number of tiles the manufacturer must make in order for the bathroom floor to be fully paved?
I have tried a few cases by keep getting problems around the edges/corners of the room. Am I correct in thinking that such an arrangement of tiles is impossible, and if so, how to formalise this?
Basically we have $10$cm left over on the $3.7$m side, and $20$cm left over on the $1.4$m side, but this leaves open a lot of possibilities...
In reality the corners would not be a problem and the manufacturer would cut some (either $1, 2$ or $4$) of the tiles twice, so I guess this problem is more of a hypothetical one.