A courtyard is 18 m 72 cm long and 13 m 20 cm wide. It is to be paved with equal square tiles of the largest size that will fit a whole number of times along both the length and the width, with no tile cut. How many tiles are needed?

In centimetres the courtyard is 1872 by 1320, so the largest usable tile has a side equal to their highest common factor: $1872=2^{4}x{3}^{2}x$ 13 and $1320=2^{3}x3\times 5\times$ 11, giving a common factor of $2^{3}x$ 3 = 24 cm. That lays 1872/24 = 78 tiles along the length and 1320/24 = 55 across the width, so $78\times 55=4290$ tiles. The trap 17160 uses a 12 cm tile — a common factor, but not the largest, so it needs four times as many tiles; 102960 divides the courtyard's area by the tile's SIDE of 24 cm instead of by its area of 576 sq cm; 2145 halves the correct count.