Problem:
A rectangular courtyard is 18 meters and 72 centimeters long and 13 meters and 20 centimeters broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.

Solution:

To find the least possible number of tiles required to pave the courtyard, we need to determine the dimensions of the tiles that would evenly cover the area of the courtyard.

Converting the measurements:
First, let's convert the measurements into a uniform format. We convert 72 centimeters into meters by dividing it by 100. Thus, the courtyard's length becomes 18.72 meters, and the breadth becomes 13.20 meters.

Finding the dimensions of the tiles:
To find the dimensions of the tiles, we can calculate the greatest common divisor (GCD) of the length and breadth of the courtyard. The GCD will give us the length of the side of each tile.

The GCD of 18.72 and 13.20 can be found by using the Euclidean algorithm. We divide the longer length (18.72) by the shorter length (13.20) and find the remainder. Then, we divide the divisor (13.20) by the remainder and repeat this process until the remainder becomes zero. The last non-zero remainder is the GCD.

Using this method, we find that the GCD of 18.72 and 13.20 is 1.44 meters.

Calculating the number of tiles:
To calculate the number of tiles required, we divide the area of the courtyard by the area of each tile.

The area of the courtyard is the product of its length and breadth: 18.72 * 13.20 = 246.144 square meters.

The area of each tile is the square of the side length: 1.44 * 1.44 = 2.0736 square meters.

To find the number of tiles, we divide the area of the courtyard by the area of each tile: 246.144 / 2.0736 = 118.8.

Rounding up the number of tiles:
Since we cannot have a fraction of a tile, we need to round up the number of tiles to the nearest whole number. Therefore, the least possible number of tiles required is 119.

Summary:
To pave the rectangular courtyard, we will need a minimum of 119 square tiles of the same size. The dimensions of each tile are determined by finding the greatest common divisor of the length and breadth of the courtyard. The GCD of 18.72 and 13.20 is 1.44 meters. By calculating the area of the courtyard and the area of each tile, we find that the number of tiles required is 118.8, which is rounded up to 119.