The largest size of tile which could be used for the purpose is 21 cm × 21 cm.

Explanation

For this purpose, let’s calculate in integers.

First, let’s convert the courtyard side lengths to cm.

• 3.78 metres long and 5.25 metres wide equals to:
  378 cm long and 525 cm wide.

Second, because the tiles are all in square shape, let’s find the greatest common factor/divisor of 378 and 525.

• 378 = 2 × 3³ × 7
• 525 = 3 × 5² × 7

gcf(378, 525) = 3 × 7 = 21

Therefore, the largest size of tile which could be used for the purpose is 21 cm × 21 cm.

The “formation” of the tiles should be (378/21) by (525/21) = 18 by 25 tiles.

CONCLUSION

∴  The largest size of tile which could be used for the purpose is 21 cm × 21 cm.