Bathroom dimensions:
- Length: 10 feet
- Breadth: 8 feet

Tile size:
To find the largest possible size of the square tile, we need to determine the greatest common divisor (GCD) of the length and breadth of the bathroom.
- GCD(10, 8) = 2

The GCD of 10 and 8 is 2, which means the largest square tile that can fit perfectly in the bathroom has a side length of 2 feet.

Number of tiles required:
To calculate the number of tiles required, we need to divide the total area of the bathroom by the area of each tile.
- Total area of the bathroom = length × breadth = 10 feet × 8 feet = 80 square feet
- Area of each tile = side length × side length = 2 feet × 2 feet = 4 square feet

To find the number of tiles required, we divide the total area of the bathroom by the area of each tile.
- Number of tiles required = Total area of the bathroom / Area of each tile = 80 square feet / 4 square feet = 20 tiles

Therefore, 20 square tiles of size 2 feet by 2 feet are required to fit the bathroom.

Size of the tile required to be cut:
Since the side length of the square tile is 2 feet, and the length and breadth of the bathroom are 10 feet and 8 feet respectively, some tiles will need to be cut to fit the remaining space.

To determine the size of the tile that needs to be cut, we need to find the remaining length and breadth after fitting as many complete tiles as possible.

- Remaining length = length of the bathroom % side length of the tile = 10 feet % 2 feet = 0 feet
- Remaining breadth = breadth of the bathroom % side length of the tile = 8 feet % 2 feet = 0 feet

Since the remaining length and breadth are both 0 feet, it means that all the tiles fit perfectly without requiring any cuts. Therefore, no tiles need to be cut.

Summary:
- The largest possible size of the square tile is 2 feet by 2 feet.
- The number of tiles required is 20.
- No tiles need to be cut as they all fit perfectly in the bathroom.